Developing a QSPR model for Alzheimer’s drugs using topological indices and M-polynomial: A computational study

Abstract

Topological indices, which are numerical descriptors that encode molecular structure, are widely used in computational drug discovery due to their efficiency and interpretability. In this study, we developed a robust quantitative structure–property relationship (QSPR) framework to predict the core physicochemical properties of nine clinically relevant Alzheimer’s disease drugs, including Donepezil, Galantamine, and Memantine.​ We employed a streamlined computational approach, using MATLAB and the M-polynomial method, to efficiently calculate a series of degree-based topological indices. Through comprehensive regression analyses, we identified strong correlations between degree-based topological indices and key physicochemical properties, including boiling point and molar refractivity. While linear models provided a reasonable baseline, nonlinear models, particularly cubic and power equations, delivered significantly improved predictive accuracy. The analysis highlighted the critical interplay between the choice of the index and the regression model. For instance, the cubic model was frequently the most effective for predicting properties such as boiling point and flash point, while the power model performed best for molar refractivity and polarizability. Notably, the redefined first Zagreb index and the modified first Zagreb index exhibited exceptional predictive capacity, reflecting their sensitivity to structural features that govern physicochemical behavior. The strong performance of these QSPR models underscores their potential to accelerate the rational design of Alzheimer’s therapeutics. By enabling rapid, cost-effective, and reliable property prediction prior to synthesis, this framework offers a valuable tool for future drug development efforts.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-31486-0.

Introduction

Alzheimer’s disease is a neurodegenerative disorder with an insidious onset and a progressively worsening course. It accounts for roughly 60–70% of dementia cases worldwide. The most rampant initial symptoms are forgetting recent events and short-term memory problems, as the disease affects the brain’s nerve function and communication, leading to the gradual destruction of brain function^1,2. In the advanced stages, the disease’s impact becomes more severe. Patients often become fully dependent on caregivers for activities of daily living and may develop aphasia, akinesia, or severe motor impairment. Accompanying these physical declines are profound psychological effects, including depression, apathy, and fatigue^3,4. Alzheimer’s is more likely to occur in women and people over the age of 65⁵. Current statistics indicate that approximately 50 million people worldwide suffer from Alzheimer’s and other forms of dementia, with projections suggesting this number could exceed 150 million by 2050^6,7.

The cause of most Alzheimer’s cases is still largely unknown, except for the 1 to 2% of cases in which determinant genetic differences have been identified. At present, the treatment of Alzheimer’s disease is focused on managing symptoms rather than providing a cure. Therapeutic strategies include symptomatic treatments, management of behavioral disorders, and drugs designed to slow the progression of the disease. While no curative therapy is yet available, several approved drugs can effectively delay disease progression and attenuate memory impairment and behavioral disturbances in some patients⁸.

Computational chemistry has emerged as a cornerstone of modern drug discovery, providing systematic frameworks to address the complexities of multifactorial pathologies such as Alzheimer’s disease. Within this domain, Quantitative Structure-Activity/Property Relationship (QSAR/QSPR) modeling represents a compelling strategy. These models establish a predictive link between a molecule’s structure and its biological or physicochemical function. By using mathematical descriptors to encode chemical information numerically, QSAR/QSPR enables the rapid estimation of a compound’s properties, thereby accelerating the identification and optimization of potential new drugs^9,10.

Topological indices are numerical descriptors derived from a molecule’s graph representation and encode critical structural information, such as branching, size, and atom connectivity. By distilling these complex structural features into single, quantitative values, they provide a powerful basis for predicting a compound’s physicochemical behavior and biological activity within QSAR and QSPR frameworks¹¹. This approach is particularly valuable in Alzheimer’s drug discovery. The process of screening vast chemical libraries for potential therapeutic candidates is a significant logistical and financial bottleneck in the development of new treatments. By enabling rapid, computational prediction of a compound’s properties, topological indices offer an efficient strategy to prioritize candidates and streamline the discovery process.

By establishing a mathematical correlation between a molecule’s structure and its properties, QSPR models based on topological indices can predict key attributes of potential Alzheimer’s disease drugs before they are synthesized and tested experimentally. This predictive capability extends to a range of critical physicochemical properties, including molecular weight, boiling point, and polar surface area, as well as to indicators of therapeutic efficacy. The application of this predictive approach offers significant advantages for drug discovery. It not only accelerates the identification of promising drug candidates but also facilitates the rational design and optimization of novel therapeutics for Alzheimer’s disease. Ultimately, integrating these computational models into the development pipeline saves considerable time and resources, streamlining the path from initial concept to clinical application^12,13.

Topological indices are established as cost-effective and powerful descriptors in chemoinformatics, particularly for estimating the physicochemical properties of small-molecule drugs^14–16. Indeed, QSPR models built upon these indices have been successfully applied to predict key molecular attributes for a wide range of therapeutic agents, including those targeting cancer, hypertension, and cardiovascular diseases. This proven utility motivates the application of topological indices to characterize the structurally diverse compounds used in the management of Alzheimer’s disease^17,18.

The utility of topological indices in QSPR modeling is well documented across diverse scientific domains. These numerical descriptors have been successfully employed to predict key properties in fields from materials science, such as characterizing boron-based nanomaterials, to chemoinformatics and nanotechnology^19–21. Furthermore, within pharmacology, topological indices have proven instrumental for modeling compounds targeting a wide array of diseases, including viral infections and various forms of cancer^22–24. This broad applicability underscores the immense potential of QSPR models to transform and expedite preclinical research. By providing a cost-effective, rapid framework for screening and optimizing chemical structures, these computational models reduce reliance on time-consuming, expensive laboratory experiments. This acceleration of the research and development pipeline is a critical advantage in the quest for novel therapeutics^25–27.

Recent advancements in the field have demonstrated that integrating topological indices with machine learning models, such as neural networks and random forests, can significantly enhance the predictive accuracy for key molecular properties²⁸. This synergy between classic descriptors and modern algorithms represents a promising frontier for developing more sophisticated and precise QSPR models.

Concurrently, other lines of research have focused on developing novel descriptors. For instance, newly proposed topological indices have been used to effectively model the structure-activity relationships of synthesized amide derivatives, demonstrating their potential as valuable tools in the search for anti-Alzheimer’s agents²⁹.

Alongside these advanced computational methods, studies continue to affirm the value of traditional approaches. Recent work has shown that even conventional linear QSPR models can provide reliable and useful estimates of physicochemical and biological attributes for novel compounds being investigated for the treatment of Alzheimer’s disease³⁰.

This study employs the M-polynomial formalism, in conjunction with edge-partitioning techniques, to systematically compute a panel of degree-based topological indices for the selected Alzheimer’s drugs. The M-polynomial serves as a powerful algebraic tool that elegantly encodes degree-based structural information of a molecular graph into a single bivariate polynomial function.

The primary advantage of this approach lies in its computational efficiency. Once the M-polynomial for a given molecular structure is derived, a wide array of degree-based topological indices can be obtained directly through simple algebraic and differential operations. This method thereby obviates the need for redundant, index-by-index calculations, creating a more streamlined and efficient workflow for characterizing chemical structures³¹.

In pharmacology, M-polynomial-based approaches have been instrumental in analyzing the physicochemical properties and structural features of various therapeutic agents. This method has been successfully used to model antiviral drugs targeting pathogens such as SARS-CoV-2, as well as anti-tumor agents such as cyclodextrin derivatives^32–34. Furthermore, polynomial mathematical models are widely used to optimize the development of nanotechnology-based products, including nanoemulsions and nanoparticles^35–40.

The broad utility of these techniques for predicting the properties of drugs and nanostructures is further evidenced^41–43. This established precedent across multiple therapeutic and formulation domains motivates our application of the M-polynomial method to systematically characterize the physicochemical attributes of drugs used in Alzheimer’s disease therapy. Therefore, this study aims to systematically investigate the relationship between molecular structure and key physicochemical properties for a curated set of nine Alzheimer’s disease drugs.

The specific compounds selected for this analysis are ABT-089, ABT-126, Donepezil, Galantamine, Memantine, Metrifonate, Phencyclidine, Rivastigmine, and Tacrine⁴⁴. The molecular structures of these compounds are depicted in Fig. 1.

Fig. 1.

Chemical structures of Alzheimer’s drugs from ChemSpider⁴⁵.

By leveraging a panel of degree-based topological indices (summarized in Table 1) derived via the M-polynomial method, we develop and validate a series of QSPR models. The central goal is to elucidate which structural features and topological descriptors are most predictive of key physicochemical attributes, thereby providing a robust computational framework to guide the rational design of future anti- Alzheimer’s disease therapeutic agents.

Table 1.

Relations of M-polynomial with some zagreb-type topological indices^47–52.

Topological index		Derivation from
First Zagreb
Second Zagreb
Third Zagreb
Redefined first Zagreb
Redefined second Zagreb
Redefined third Zagreb
Modified first Zagreb
Modified second Zagreb
Hyper first Zagreb
Hyper second Zagreb
Nano Zagreb
Augmented Zagreb

Methodology

This study centers on a set of nine previously mentioned drugs relevant to Alzheimer’s disease. The two-dimensional molecular graphs for these compounds, shown in Fig. 1, were used as the basis for all subsequent graph-theoretical analyses. Suppose that is a simple connected graph representing a molecular structure where is the set of vertices (atoms) and is the set of edges (bonds).

Two vertices and in are called adjacent if is a member of the set of edges of . The number of vertices adjacent to a vertex is called the degree of , which is denoted by The foundation for constructing the M-polynomial is the edge-partitioning method. This technique involves classifying the edges of the molecular graph into distinct sets based on the degrees of their endpoint vertices. For a graph , an edge belongs to the partition if the degree of the vertex s and the degree of the vertex is .

By systematically partitioning all edges in this manner, we can accurately construct the M-polynomial, which then acts as a generator for the desired topological indices⁴⁶. The M-polynomial of a graph is formally defined as:

1

where is the number of edges such that ⁴⁷.

This combined approach provides a rigorous, efficient, and transparent framework for analyzing the structural properties of complex molecules. Its systematic nature makes it particularly well-suited for the comparative analysis of the Alzheimer’s drugs investigated in this paper.

Table 1 shows Zagreb-type topological indices and their derivation from the M-polynomial , in which

The following 5-step algorithm was implemented to compute and analyze the topological indices defined in Table 1 using the M-polynomial method. Subsequently, this framework is used to develop and present a QSPR model that examines the correlation between these indices and the physicochemical properties of the selected Alzheimer’s drugs.

Step 1: Draw and calculate its adjacency matrix using TopoCluj.

Step 2: Determine the cardinality matrix for the edge-partition method using our MATLAB program to input the matrix .

Step 3: Use our MATLAB program to input a matrix and compute M-polynomial for Alzheimer’s drugs.

Step 4: Compute Zagreb-type topological indices in Table 1 using our MATLAB code.

Step 5: Employ SPSS software to generate regression models and select the most appropriate model.

The pseudo-code of the proposed algorithm is presented in Appendix A.

Calculation of the M-polynomial of alzheimer’s drugs using MATLAB programming

This section details our main computational results. We utilize the M-polynomials to compute several topological indices. We analyze the molecular structures of Alzheimer’s drugs, represented as graphs in Fig. 1. By examining these molecular graphs, we extract the necessary information about the structural characteristics of these drugs to input into our MATLAB programming.

• ABT-089: Let be the molecular graph of the ABT-089 drug with the molecular formula. This graph has order 14 and size 15 with and . We input the edge-partition matrix of the graph as in our MATLAB.

• ABT-126: Suppose that is the graph of the molecular structure of the ABT-126 drug with the molecular formula. This graph contains 11 vertices, 10 edges, and . The edge-partition matrix of the graph is as .

• Donepezil: The molecular graph of the drug Donepezil is denoted as , and it has the molecular formula. This graph contains 28 vertices and 31 edges with and . Also, we have .

• Galantamine: Let be the molecular graph of the Galantamine drug with the molecular formula. This graph has 22 vertices and 25 edges with and . We obtain the edge-partition matrix of the graph as .

• Memantine: We suppose that is the molecular graph of the Memantine drug with the molecular formula. This graph has 13 vertices and 15 edges with and . Furthermore, we obtain .

• Metrifonate: Let be the molecular graph of the Metrifonate drug with the molecular formula. This graph has 12 vertices and 11 edges with and . We input the matrix in our MATLAB programming.

• Phencyclidine: Let be the molecular graph of the Phencyclidine drug with the molecular formula. This graph has 18 vertices and 20 edges with and . Also, we have .

• Rivastigmine: Let be the molecular graph of the Rivastigmine drug with the molecular formula. The graph has 17 vertices and 17 edges with and . The edge-partition matrix of the graph is as .

• Tacrine: Let be the molecular graph of the Tacrine drug with the molecular formula. This graph has 15 vertices and 17 edges, which and . We input the edge-partition matrix of the graph as in the MATLAB code.

The following theorem obtains the M-polynomial expression for these drugs.

Theorem 1

For the molecular graphs of the Alzheimer’s drugs described above,

i)

ii)

iii)

iv)

v)

vi)

vii)

viii)

ix)

Proof

i) Since , for the molecular graph of the ABT-089 drug, we have , , , and . Therefore, using the definition (1), we get

Similar to the proof of the case , the remaining cases are proved using relation (1) and the obtained edge-partition matrix . ■

Illustrative example: calculation for the ABT-089 drug

In this section, we provide a detailed, step-by-step manual calculation of the twelve Zagreb-type topological indices listed in Table 1 of the ABT-089 drug. These calculations verify the results from our MATLAB code and illustrate the practical application of the M-polynomial method. The molecular graph of ABT-089, denoted as , has the following M-polynomial, which was derived in Theorem 1. The M-polynomial for the molecular graph of ABT-089 () is:

To compute the topological indices, we use the operators defined in Table 1, where . The key operators are:

Below are the manual calculations for each of the 12 topological indices.

• First Zagreb index:

The formula is . Therefore, we get.

Evaluating at :

• Second Zagreb Index: The formula is . Therefore, we get

  

  

  

Evaluating at :

• Third Zagreb Index: The formula is . Therefore, we get

  

Evaluating at :

• Redefined First Zagreb Index: The formula is . Therefore, we have

  

Evaluating at :

• Redefined Second Zagreb Index:

• The formula is . Consider,

Evaluating at :

• Redefined Third Zagreb Index:

The formula is . Let.

Therefore, we get

Evaluating at :

• Modified First Zagreb Index:

The formula is .

Evaluating at :

• Modified Second Zagreb Index:

The formula is . We have

Evaluating at :

• Hyper First Zagreb Index:

The formula is . Suppose that

Evaluating at x = 1:

• Hyper Second Zagreb Index:

The formula is . We have

And consequently,

Evaluating at :

• Nano Zagreb Index:

The formula is . We get

hence

Evaluating at :

• Augmented Zagreb Index:

The formula is . Consider,

Therefore, we get

Consequently,

Evaluating at :

The calculated values match the numerical results presented in the manuscript, confirming the accuracy of both the derived formulas and the computational algorithm.

Numerical results and regression models

We employ the provided MATLAB code to compute the Zagreb-type topological indices defined in Table 1 using Theorem 1. As an example, we calculate the topological indices of the ABT-089 graph using our MATLAB code. The resulting values are presented below.

Output of our MATLAB programming for the ABT-089 drug:

The order = 14.

The size = 15.

The maximum degree = 3.

The minimum degree = 1.

The first Zagreb index is 28*x^2*y^2 + 30*x^2*y^3 + 6*x^3*y^3 + 4*x*y^3.

Z1(G) = 68.

The second Zagreb index is 28*x^2*y^2 + 36*x^2*y^3 + 9*x^3*y^3 + 3*x*y^3.

Z2(G) = 76.

The third Zagreb index is 6*x^2*y^3 + 2*x*y^3.

Z3(G) = 8.

The redefined first Zagreb index is 7*x^2*y^2 + 5*x^2*y^3 + (2*x^3*y^3)/3 + (4*x*y^3)/3.

Rz1(G) = 14.

The redefined second Zagreb index is (31*x^4)/4 + (36*x^5)/5 + (3*x^6)/2.

Rz2(G) = 329/20.

The redefined third Zagreb index is 112*x^2*y^2 + 180*x^2*y^3 + 54*x^3*y^3 + 12*x*y^3.

Rz3(G) = 358.

The modified first Zagreb index is 2*x^4 + (6*x^5)/5 + x^6/6.

M1*(G) = 101/30.

The modified second Zagreb index is (7*x^2*y^2)/4 + x^2*y^3 + (x^3*y^3)/9 + (x*y^3)/3.

M2*(G) = 115/36.

The Hyper first Zagreb index is 128*x^4 + 150*x^5 + 36*x^6.

HM1(G) = 314.

The Hyper second Zagreb index is 112*x^2*y^2 + 216*x^2*y^3 + 81*x^3*y^3 + 9*x*y^3.

HM2(G) = 418.

The Nano Zagreb index is 30*x^2*y^3 + 8*x*y^3.

NZ(G) = 38.

The Augmented Zagreb index is Az(G)=:(475*x^2)/8 + 48*x^3 + (729*x^4)/64.

Az(G) = 7601/64.

M-polynomial is 7*x^2*y^2 + 6*x^2*y^3 + x^3*y^3 + x*y^3.

The numerical results for the topological indices of Alzheimer’s drugs are stated in Table 2. We employ various regression models to analyze the QSPR model, focusing on the topological indices computed from the molecular structures of Alzheimer’s drugs. The regression equations considered in this study are as follows.

The linear equation:

Quadratic equation:

Cubic equation:

Logarithmic equation:

Inverse equation:

Power equation:

S-Curve equation:

Exponential equation:

Compound equation:

Growth equation:

where and denote the topological index and the physicochemical property, respectively. The constant and the coefficient of the regression model are represented by , , , and ⁵³.

To evaluate the predictive performance of the regression models, three error metrics were employed: Mean Absolute Error (MAE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE). These are defined as

where denotes the observed value, the predicted value, and the number of observations. MAE captures the average absolute deviation, MSE penalizes larger deviations more strongly, and RMSE expresses error in the same units as the predicted property.

In addition to error metrics, the coefficient of determination () was used to assess model performance. is defined as.

where ​ is the mean of the observed values. quantifies the proportion of variance in the observed data explained by the model, with values closer to 1 indicating a better fit⁵⁴.

Data analysis and discussion

This section evaluates the predictive power of the topological indices listed in Table 1 for a set of Alzheimer’s drugs. The goal is to build a QSPR model. To do this, we correlate the calculated topological indices (Table 2) with key experimental physicochemical properties (Table 3). These properties, including boiling point (BP), enthalpy of vaporization (E), flash point (FP), molar refractivity (MR), molar volume (MV), polarizability (P), and polar surface area (PSA) were selected because they are critical for determining a drug’s pharmacokinetic behavior. For instance, PSA is a crucial factor for predicting a drug’s ability to cross the blood-brain barrier, an essential requirement for any Alzheimer’s therapeutic. Similarly, MR and MV relate to the molecule’s size and binding potential. By establishing strong regression models between these properties and our indices, we aim to demonstrate that molecular structure, as captured by topological indices, can effectively predict a compound’s real-world behavior. The predictive performance of the regression models was assessed using and three error metrics: MAE, MSE, and RMSE. Lower values of MAE, MSE, and RMSE indicate better predictive accuracy, while higher values indicate a greater proportion of the observed variance that the model explains. The best-performing model is therefore identified as the one with the highest and lowest error metrics, with a significance level of less than .

Table 2.

The numerical values of Zagreb indices of Alzheimer’s drugs.

Drugs
ABT-089
ABT-126
Donepezil
Galantamine
Memantine
Metrifonate
Phencyclidine
Rivastigmine
Tacrine

Table 3.

The physicochemical features of alzheimer’s drugs.

Drugs	Flash point (FP)	Boiling point (BP)	Polarizability (P)	Enthalpy of vaporization (E)	Polar surface area (PSA)	Molar Refractivity (MR)	Molar Volume (MV)
ABT-089
ABT-126
Donepezil
Galantamine
Memantine
Metrifonate
Phencyclidine
Rivastigmine
Tacrine

Interpretation of key correlations

Table 4. The correlation coefficient calculated by linear regression models between Zagreb indices and the physicochemical properties of Alzheimer’s drugs∙.

Table 4.

Presents the correlation coefficients (R) between the topological descriptors and the physicochemical properties of the drugs, derived from linear regression equations. The results indicate that some indices, particularly, , and , show a very high correlation with most of the studied properties (except for PSA). In contrast, indices such as , , and do not show significant correlations. Notably, the PSA property shows no significant relationship with any of the topological indices in the linear model.

Topological index

The detailed linear regression models that generated these high correlation values, including coefficients, R-squared values, and statistical significance, are presented in Tables 5 and 6.

Table 5.

The linear regression equations that give the best approximation for the physicochemical properties FP, BP, and P of alzheimer’s drugs.

Regression equations		Std. Error	MAE	MSE	RMSE	P-value

Table 6.

The linear regression equations that give the best approximation for the physicochemical properties E, MR, and MV of alzheimer’s drugs.

Regression equations		Std. Error	MAE	MSE	RMSE	P-value

A detailed examination of our regression models, presented in Tables 4 and 5, and 6, indicates that specific topological indices are particularly effective at predicting certain physicochemical properties. The analysis of correlation coefficients (R) in Table 4 reveals a striking pattern: demonstrates exceptionally strong correlations across a majority of the properties, including FP (), BP (), P (), and MR ().

The strength of these correlations is confirmed by the linear regression models detailed in Tables 5 and 6. For example, the linear regression model using demonstrated superior performance in predicting MR. The model, defined by the equation, achieved a coefficient of determination () of , indicating it accounts for of the variance in MR. The model’s high accuracy is reinforced by low error metrics, including an MAE of 3.18, an MSE of 13.01, and an RMSE of . Similarly, the linear model for P, defined by the equation, yielded a high coefficient of determination () and minimal errors (, , ). The superior performance of the index is mechanistically significant. As a descriptor defined by the sum of connectivity ratios, it is susceptible to branching in the molecular structure. It is a strong correlation with properties like Molar Volume () and Enthalpy of vaporization () suggests that intermolecular forces are heavily influenced by the molecule’s specific topology, not just its size. Furthermore, other indices show strong predictive power. As shown in Table 4, the modified Zagreb indices, , and , also exhibit excellent correlations, particularly with BP ( for ) and MR ( for ). The corresponding linear model for BP using () is one of the most robust, with an of and low error metrics (, , ) (Table 5). Also, the linear model for FP using () is highly significant, as its value of indicates that this index alone can explain over of the variance in flash point supported by low errors ( ).

This underscores the importance of inverse-degree-based descriptors, which effectively capture the contributions of atoms with lower connectivity, often found in peripheral functional groups.

In contrast, no significant correlations were found for , , and indices. This suggests that descriptors based solely on the difference in vertex degrees, such as and , are less informative for predicting these bulk physicochemical properties.

Critically, PSA, a key parameter for drug absorption and blood-brain barrier penetration, shows no significant linear correlation with any of the studied indices (Table 4). This highlights a limitation of these 2D topological descriptors: while excellent at capturing properties related to overall molecular size and branching, they may be insufficient for predicting properties governed by specific polar-atom arrangements.

Moving beyond linear relationships, our analysis of non-linear models revealed even stronger and more nuanced predictive capabilities. The detailed results for all non-linear regression models, including quadratic, cubic, logarithmic, inverse, power, S-curve, exponential, compound, and growth equations, are provided in Appendix B (Tables B1-B9).

A key finding is the remarkable performance of , which consistently emerged as one of the most powerful predictors across multiple nonlinear models for properties such as MR and P.

For instance, using a power model, predicts MR with an of (Table B5), and using a cubic model, it predicts P with an of (Table B2). The widespread success of is particularly noteworthy. As a reformulated Zagreb-type index, it is susceptible to the distribution of edge degrees across the molecular graph, effectively capturing complex structural features, such as the presence of heteroatoms and branching patterns. Its dominant performance suggests that these specific structural characteristics are the primary drivers for properties that depend on electron distribution and molecular volume.

However, our analysis also identified specific cases where other indices provided superior predictions, highlighting the importance of selecting the right descriptor for the right property. For instance, in the cubic regression model, the index for predicting FP and BP, and the index for predicting E, were exceptionally accurate. For FP and BP, the cubic model based on yielded the highest coefficient of determination with and , respectively (Table B2). For E, the cubic model using the index was the most powerful predictor, with an value of (Table B2).

The success of different indices in predicting specific properties is chemically intuitive. Properties like FP and E are highly dependent on complex intermolecular forces that are better captured by the intricate formulations of the and indices, whereas properties like MR and P are more directly related to the overall connectivity and branching captured by

Furthermore, logarithmic models revealed unique relationships for certain properties. While not always the top-performing model, they offer insight into non-linear saturation effects. The index proved to be the best predictor for E within a logarithmic framework (, Table B3), while the index was superior for estimating MV (, Table B3). The strong performance of a logarithmic model for these properties suggests that, as molecular complexity increases, their values do not grow linearly but rather approach a plateau. This implies that beyond a certain molecular size or complexity, adding more atoms yields diminishing returns in increasing molar volume or enthalpy of vaporization, a phenomenon well captured by a logarithmic function.

Significance of model type and performance

A central finding of our comparative analysis, summarized in Table 7., is that the choice of regression model is as critical as the selection of the topological index itself. While linear models provide a helpful baseline, our results consistently show that non-linear models offer superior accuracy for predicting the physicochemical properties of the studied Alzheimer’s drugs. This is evidenced not only by higher values but also by correspondingly lower error metrics (MAE, MSE, and RMSE) across the non-linear models. To visually supplement these statistical findings, Fig. 2 presents the regression plots for the most significant correlations. These graphs are essential for comparing the goodness-of-fit of various model types. By illustrating the distribution of data points around the regression curves, the plots provide an intuitive understanding of each model’s predictive power. Specifically, they highlight instances where non-linear models (e.g., cubic, power) more accurately capture the underlying structure-property relationships than simple linear models, thereby visually reinforcing our conclusion that non-linear approaches offer superior accuracy for this dataset.

Table 7.

Choosing the best equation to estimate the physicochemical properties of alzheimer’s drugs with topological indices using max .

Property	Indices	Equations
		Linear	Quadratic	Cubic	Logarithmic	Inverse	Power	S-Curve	Exponential	Compound	Growth

Fig. 2.

Graphical representation of regression models between topological indices and physicochemical properties of drugs.

The cubic model, while often more complex, demonstrated a unique and widespread advantage in our study. It emerged as the best predictor for FP and BP when using indices such as , , , , , , , , and . It was also the top-performing model for predicting the Enthalpy of E with all indices except and . This suggests that these specific properties are governed by highly complex, non-linear interactions that are only adequately captured by a higher-order polynomial.

The power model also demonstrated exceptional utility, emerging as the best predictor for properties such as MR and P across several topological indices, including, , , , and . This strong performance suggests that the relationship between these structural descriptors and specific properties is not merely additive but follows a multiplicative or power-law dependency. Such behavior is often characteristic of complex physicochemical phenomena, where properties scale non-linearly with molecular attributes, such as size and electron distribution.

Interestingly, for several properties, different model types yielded nearly identical high-accuracy predictions. For instance, for FP and BP prediction using the index, both quadratic and cubic models achieved the same highest value (). This redundancy implies that, for these specific structure-property relationships, adding a third-order (cubic) polynomial term does not capture significantly more variance than a simpler second-order (quadratic) model. In such cases, the quadratic model would be preferred for its greater parsimony. A similar pattern of shared high performance was observed among the exponential, compound, and growth models for predicting MV, with the,, and indices, indicating that an exponential growth function robustly describes this particular relationship. Given their mathematical equivalence, these models inherently produced identical fits and R-squared values, confirming the suitability of a single exponential model for reporting.

In contrast to other models, the logarithmic model did not emerge as the top predictor for any property in this study. While such models can be insightful for phenomena involving saturation, our findings indicate that for this dataset, other non-linear functions provide a better fit.

In conclusion, our analysis extends beyond a simple ranking of indices, demonstrating that a nuanced, property-specific approach to model selection is essential. The findings provide a clear roadmap: for properties governed by complex interactions, such as FP, BP, and E, higher-order polynomials, like the cubic model, are often necessary. For properties governed by scaling laws such as MR and P, power models are superior. This understanding is critical for building truly predictive QSPR frameworks in the future.

A primary limitation of this study is the small sample size () available for the regression analysis. This constrains the statistical robustness of our models and is a common challenge in QSPR studies of specific drug classes, such as those for Alzheimer’s disease, where the number of approved or late-stage compounds is inherently limited^9,12,55. Consequently, our findings should be interpreted as exploratory and hypothesis-generating. Although the chemical homogeneity of our curated dataset may partially mitigate variance, we recommend that these models be validated with larger and more diverse datasets in future studies to confirm their predictive power and generalizability.

Conclusion

This study successfully developed robust QSPR models to predict the physicochemical properties of nine key Alzheimer’s drugs. The significant findings are summarized as follows:

• Zagreb-type topological indices, efficiently calculated using the M-polynomial method, served as effective molecular descriptors for building the predictive models.

• Nonlinear regression models, particularly cubic and power equations, demonstrated significantly greater predictive accuracy than standard linear models.

• The selection of an optimal regression model proved to be property-specific, highlighting that the predictive power of a topological index is maximized only when paired with the appropriate mathematical model.

• The cubic model was most effective for properties governed by complex intermolecular forces, such as BP and E. Indices like and yielded outstanding coefficients of determination () up to 0.947 and 0.922, respectively.

• The power model was the ideal choice for properties related to electron distribution and molecular volume, such as MR and P. Within this model, the index achieved an exceptional value of 0.963 for MR.

While our models show strong predictive capability, this work opens several promising avenues for future research. A primary limitation of this study is the small sample size (), which is a common challenge for specific drug classes. Therefore, a critical next step is to validate these models on larger, more diverse datasets to ensure their generalizability. Future work should also extend this QSPR framework to a QSAR analysis to correlate these indices with biological activities, incorporate 3D descriptors to capture spatial features more effectively, and apply machine learning algorithms to uncover more complex patterns for the design of next-generation Alzheimer’s therapeutics.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

M.H.A. and M.H.S. conceptualized the study and supervised the research process. F.M. and M.Z. performed the computational analyses, including MATLAB coding and SPSS modeling. F.M. also contributed to data collection and graph analysis. All authors participated in the interpretation of results, contributed to the writing and critical revision of the manuscript, and approved the final version.

Data availability

Data, including all molecular structures and properties, is available at http://www.chemspider.com. The authors provide the MATLAB code upon request.

Declarations

Competing interests

The authors declare no competing interests.