Introduction

Chemotherapy, a fundamental aspect of medical oncology, involves usage of one or more anti-cancer drugs in a standard regimen to treat cancer1. It targets rapidly dividing cancer cells, damaging their DNA and preventing replication, which leads to cell death2,3. Unlike antibiotics, chemotherapy drugs integrate with DNA, inhibiting RNA synthesis and breaking down DNA strands4. The ongoing development of predictive models is crucial for discovering effective therapeutic drugs. There are several major problems with current chemotherapy therapies, making the development of new medications an absolute necessity. Severe side effects, including harm to healthy cells, are common with current chemotherapy medications and might manifest as nausea, exhaustion, and hair loss5. In addition, the effectiveness of treatment decreases with time since some tumours develop resistance to current medications6. Furthermore, due to the heterogeneity of cancer kinds, it is necessary to develop medicines that can zero in on certain biochemical pathways7. Therefore, there is a constant demand for novel therapeutic approaches to enhance treatment effectiveness and patient well-being while reducing side effects. In light of these obstacles, it is critical to seek novel chemotherapeutic medications to improve cancer treatment results8,9.

Chemical graph theory is a branch of mathematical chemistry that models chemical structures using graphical techniques10. It involves representing molecules as graphs, where atoms and bonds represent vertices and edges, respectively. This method facilitates the examination of molecular structure, connectivity, and characteristics11. In chemical graph theory, topological indices play a crucial role as numerical descriptors that include the structural aspects of molecules in graph-based representations12. Predicting a wide range of physical, chemical, and biological features is made possible by these indices, which store structural and topological information about molecules13,14,15,16,17. Researchers frequently use topological indices to analyze the effects of molecular characteristics on experimental outcomes18. This paper is significant for medicinal chemistry and computational drug design researchers, as it provides a detailed analysis of how topological indices can enhance the drug discovery process19. The study explores the process by which QSPR models can use chemical structure-derived topological indices to forecast octane isomer characteristics20. In their study, Öztürk et al.21 determined topological indices for commutative ring prime ideal sum graphs. In order to forecast the physicochemical properties of cardiovascular medications, Arockiaraj et al.22 employed modified reverse degree topological indices. For the intricate structure of ruthenium bipyridine, Abirami23 calculated the degree-based topological indices. When estimating the boiling point of benzenoid hydrocarbons, the first hyper-Zagreb index comes out as best estimated24. Quantitative structure-activity relationships (QSAR) and quantitative structure-property relationships (QSPR) employ topological indices generated from chemical graphs to predict the biological activity or qualities of molecules25.

The prediction of the properties of a chemical compound can be made through investigations of quantitative structure-property relationships (QSPR) based on their molecular structure. Previous studies represent several types of research, regarding the importance of the QSPR model in predicting drug activity. Topological index and other mathematical methods are used to measure the behavior of organic compounds, including biological activities, has become a major subject in various fields of study. They compared the performance of the QSPR model with previous studies and found that the model using topological parameters was more accurate in predicting chemical compounds. These strategies involve developing mathematical models that correlate the structural attributes of molecules including drug structures with their physicochemical properties26. Integrating diverse topological indices and advanced modelling methods, such as multigraph representations and linear regression, has enhanced our understanding of how molecular structure relates to drug properties. The potential of QSPR to further the discovery and optimization of new drug candidates, as well as the development of safer and more effective therapeutic agents, is a key factor in this advancement.

For instance, Parveen et al.27 explored the importance of using degree based topological indices in QSPR models to predict fungicide properties. They used several topological descriptors and correlated them with the physicochemical properties of the fungicides. To forecast melting point of 349 organic molecules, Tahir et al.28 created a QSPR analysis using molecular descriptors based on linear regression modelling. Wei et al.29 studied QSPR through various topological indices on a variety of medication chemical structures. The author used these techniques to predict how chemical structures relate to their properties. Ahmed et al.30 predicted the physical and biological properties of novel Alzheimer’s disease through QSPR analysis based on linear regression modeling. Topological indices were used in the QSPR study of several significant COVID-19 medicines by Ozturk et al.31. The author investigated the structure of several drug candidates for COVID-19 through topological coincides and compared them with their physicochemical properties. Zaman et al. used degree-based topological discriptors and linear regression modeling to investigate new medications for treatments of blood cancer32. To evaluate the physicochemical features and anti-hepatitis medications, Mahboob et al.33 used QSPR analysis in conjunction with linear regression models. Zhang et al. utilized QSPR analysis with topological indices to study drugs for schizophrenia treatment34. This approach facilitates the selection process by enabling researchers to evaluate drug candidates based on specific criteria, leading to more efficient and informed decision-making.

Density Functional Theory (DFT) gives accurate information about the electronic structure, which makes the descriptors used in QSPR models more reliable. Thomas35 and Fermi36 presented the inaugural density functional approximation. The simplifications of the Hartree–Fock37,38 approach by Slater39 have facilitated realistic DFT computations. A QSPR model for several chlorine-substituted biphenyl systems is presented in this paper using linear and multi-linear regression analysis. The model uses conceptual DFT and information-theoretic descriptors to predict the lipophilicity (log KOW) of polychlorobiphenyl congeners40. The author investigated the stability and electrical characteristics of the PVP-CaCO3 blends using DFT computations. To further track the responsiveness of the suggested PVP-CaCO3 interactions, QSAR computations were conducted41. Using DFT analysis and QSPR modelling, the work delves into the control and prediction of defect-related characteristics in ZnO nanosheets42. By utilizing topological indices derived from the molecular structure, the study demonstrated an improvement in the accuracy of predictive curvilinear models for fibrates drug activity43. Another relevant study applied these techniques to monocarboxylic acids, examining the thermodynamic properties and their relationship with temperature-based topological indices44. These studies collectively suggest that the integration of topological indices with curvilinear regression modelling and DFT calculations can significantly enhanced prediction of drug activity, providing a robust framework for computational drug discovery and cheminformatics45.

Materials and methods

Here, we examine the efficacy of chemotherapy medications using curvilinear regression models that are based on molecular characteristics. We investigated distance-based topological indices, such as Weiner index46, Schultz index47, Harary index48, Additive Harary index49, Multiplicative Harary index50 and the Guttmann index51. These indices evaluate the performance of curvilinear regression models for predicting the action of chemotherapeutic medicines and for classifying molecular descriptors. All distance based topological indices with their respective mathematical description are shown in Table 1.

Table 1 Distance-based topological discriptors:
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The drugs used in this study, along with their DrugBank IDs, are: Gemcitabine(DB00441), Cytrabine(DB00987), Fludarabine(DB01073), and Capecitabine(DB01101) are used in the treatment of various cancers. Treatments for lymphomas, acute lymphocytic leukemia (ALL), and acute myeloid leukemia (AML) sometimes involve the use of fludarabine and cyclobine52. Capecitabine is a chemotherapy drug used to treat gastric cancer, breast cancer and colorectal cancer53. Gemcitabine is chemotherapy medication very effective in Pancreatic Cancer treatments54. The research indicated that Clofarabine(DB00631), in conjunction with other chemotherapeutic drugs, demonstrated a markedly beneficial effect in specific prognostic risk groupings of AML patients55. Altretamine(DB00488) is less hazardous than other medications employed in the treatment of refractory ovarian cancer56. Dacarbazine(DB00851) is a chemotherapeutic agent employed in the management of melanoma and Hodgkin’s lymphoma57. These drugs interfere with an enzyme called liposomal, which helps to resolve DNA during replication. By inhibiting these enzymes, topoisomerase inhibitors cause DNA damage and cell death drugs slow mitosis (cell division) by distorting microtubules, the main components of cell structure58.

Predicting many different thermodynamical features has become easier with that QSPR modelling. For these forecasts, the QSPR models makes use of topological indices. All of the following properties are taken into account: Dipole moment (DM), Zero-point vibrational energy (ZpVE), Molar entropy (S), Complexity (C), Polarisability (P), Heat capacity (CV), Topological polar surface area (TPA), Sum of electronic and zero-point energies (SEZpE), thermal (E), and Octanol-water partition (xLogp3) coefficients. To examine the connections between these attributes and the topological indices, curvilinear regression (quadratic, cubic, and linear) is used. The statistical capabilities of SPSS and MATLAB are utilised for the model to produce statistical parameters.

The DFT calculations were carried out using Material Studio59 version 8.0 from BIOVIA60 based on DMol361 optimised geometries for chemotherapy. The findings were utilised to learn more about the electron density mapped with electrostatic potential (ESPMs), density of state (DOS) plots, optimum geometries, and the energies of the lowest unoccupied and highest occupied molecular orbitals (HOMO and LUMO, respectively). Using the B3LYP (Becke, 3-parameter, Lee-Yang-Parr) hybrid functional, the DFT calculations of the drug derivatives under study were performed. In density-functional theory (DFT), hybrid functionals that incorporate additional empirical/ab initio exchange together with some Hartree-Fock exchange are used to approximate the exchange-correlation energy. The choice of the B3LYP functional with the 6-31G(d, p) basis set is likely the preference of most computational chemistry studies because it best balances accuracy and computation time. The B3LYP functional itself is basically a hybrid density functional, mixing Hartree-Fock exchange with DFT; this feature has made it very applicable to a wide class of systems. The 6-31G(d, p) basis set, which itself is also relatively small and includes the polarization of non-hydrogen atoms, is popular because it offers a fairly good balance between cost and accuracy62. A basis set [6 − 31G(d, p)] was combined with the presentation of the Hamiltonian component in the Schrödinger equation to represent the eigenvalue wave function. Two polarization basis functions are used to amplify the set: one for all hydrogen atoms and one for heavy atoms (carbon, oxygen, nitrogen, and fluorine). The double zeta (G) basis set is modest. The majority of the thermodynamical parameters of the tested chemotherapeutic derivatives, which will be described in detail in the next section, were obtained from frequency calculations carried out at the same level of optimization theory. Optimal frequency and thermodynamical calculations were performed using the software package developed by Gaussian63 in 2009. The optimised structures were visualised using GaussView (version 5.0.8)64, The Avogadro package65 was used to draw ESPMs, and GaussSumprogram66 was used to make DOS charts.

To evaluate the efficacy of chemotherapeutic medications, this part aims to build a quantitative structure-property (QSPR) link between topological indices and certain thermodynamical qualities and activities. The antiviral activity was modelled using six distance topological indicators. Investigated chemotherapeutic medicines based on geometries optimised by DMol3. The DFT calculations were carried out using BIOVIA’s Material Studio 8.0. The following quantities were calculated: molar entropy (S), electrical and zero-point energies (SEZpE), zero-point vibrational energy (ZPVE), thermal energy (E), dipole moment (DM) and heat capacity (CV). The properties with are extracted from ChemSpider67 and PubChem55 are as follows: polarisability (P), complexity (C), topological polar area (TPA), and octanol-water partition coefficients (XlogP3). Cancer treatments including Gemcitabine, Cytarabine, Fludarabine, and Capecitabine were the subjects of these computations. Curvilinear regression analysis, which may be performed using SPSS statistical software, allows for curves to be fitted instead of straight lines. Curvilinear regression models use topological indices as their independent variables, as will be shown below. Evaluations are carried out according to the formulas provided below.

$$\:y\:=\:a\:+\:bx,\:\:\:\:\:\:\:\:\:\:\text{n},\:{R}^{2},\:\text{F},\text{S}\text{e},\text{S}\text{F}\:\:\:\:\:\:\:\left(Linear\:regression\right)$$
$$\:y\:=\:a\:+{b}_{1}x\:+{b}_{2}{x}^{2},\:\:\text{n},\:{R}^{2},\:\text{F},\text{S}\text{e},\text{S}\text{F}\:\:\left(Qudratic\:regression\right)$$
$$\:y\:=\:a\:+{b}_{1}x\:+{b}_{2}{x}^{2}+{b}_{3}{x}^{3},\:\:\text{n},\:{R}^{2},\:\text{F},\text{S}\text{e},\text{S}\text{F}\:\:\left(\text{C}\text{u}\text{b}\text{i}\text{c}\:\text{r}\text{e}\text{g}\text{r}\text{e}\text{s}\text{s}\text{i}\text{o}\text{n}\right)$$

In the given equation, \(\:y\) stands for the thermodynamical attribute, \(\:x\) for the independent variable, \(\:a\) and \(\:{b}_{i}\left(i=\text{1,2},3\right)\) for the regression constants, and n for the sample number. The coefficient of correlation, abbreviated as R SE stands for Fisher’s statistics, and F for Fisher’s estimate. To forecast the model’s efficiency and compare the outcomes with experimental data, the Root Mean Square Error (RMSE) is calculated. The optimal predictive model is defined as the one with the lowest root-mean-squared error (RMSE), which is determined to be:

$$\:RMSE=\sqrt{\frac{{\sum\:}_{I=1}^{n}{\left({y}_{i}-\widehat{{y}_{i}}\right)}^{2}}{n}}$$

While \(\:n\) is the sample number, the independent variable \(\:{y}_{i}\) represents the observed values and \(\:\widehat{{y}_{i}}\) represents the predicted values. When evaluating the precision of a regression model or other predictive model, the Root Mean Square Error (RMSE) becomes crucial. One way to measure the accuracy of a prediction system is by looking at its root-mean-squared error (RMSE). It provides an idea of the degree to which the model’s forecasts agree with the observed data. A lower RMSE indicates that the model is more accurate. Researchers can use root-mean-squared error (RMSE) to validate models, pick the best model for their data, and understand how well the models fit. By analysing the prediction errors, root-mean-squared error (RMSE) reveals potential model improvement areas. In a nutshell, it gives a brief synopsis of the process for making accurate predictions, which in turn helps with comparison and, ultimately, with understanding the model fit and how to improve it.

Results and discussion

Electronic structural properties computation

The key features of the seven chemotherapy derivatives that were studied are shown in Figs. 1, 2, 3, 4, 5, 6 and 7. These features include optimized geometries, total density of states (DOS) plots, optimized electron density mapping with electrostatic potential (ESPM), and special distributions of the highest occupied molecular orbitals (HOMOs) and lowest unoccupied molecular orbitals (LUMOs). ESPMs reveal the distribution of electron densities in the seven non-planar molecules with respect to the electrostatic potentials, which in turn reveal which parts of the molecules are most vulnerable to attack from electrophilic or nucleophilic substances, depending on their electron density. Colors blue (representing a positively charged region) and red (representing a negatively charged region) indicate nucleophilic and electrophilic attack zones, respectively. The atoms that are most strongly electronegative, like Oxygen, Nitrogen, and Fluorine, are shaded a deep red; the atoms that are least strongly electronegative, Hydrogen, are shaded blue; and the atoms that are most neutral, like Carbon, are shaded white, indicating that their electronegativity is intermediate. As a result, ESPMs can pinpoint exactly where an electrophile or nucleophile is attacking a molecule. The number of possible energy levels that electrons in a system can occupy is shown by the molecule’s DOS plot. We looked at the seven HOMO energies. Destabilized HOMO, which is less negative, increases a molecule’s capacity to donate electrons. The seven derivatives’ electron-donation abilities can be organized in the following way: Dacarbazine(− 5.3204eV)\(\:>\)Altretamine(− 5.5492eV)\(\:>\)Flurabine (− 6.08065eV)\(\:>\)Cytarabine(− 6.25372eV)\(\:\:>\)Clofarabine(− 6.2667eV)\(\:>\) Gemcitabine(− 6.44284eV)\(\:\:>\)Capcitabine(− 6.513864 eV). Conversely, LUMO energy quantifies the electrons’ capacity to take up a molecule; the greater the electrons’ capacity, the more negative the LUMO. Consequently, the derivative’s electron-accepting capacity is Capcitabine (-1.67567eV) \(\:>\)Gemcitabine(− 1.131449eV) \(\:>\) Dacarbazine(− 0.9638eV)\(\:\:>\)Clofarabine(− 0.896eV)\(\:>\)Cytarabine(− 0.83049eV)\(\:>\)Flurabine(0.59375eV)\(\:\:>\)Altretamine(0.8272 eV). The HOMO and LUMO orbital energies do not distribute in the same way in Capcitabine. The molecule exhibits greater LUMO than HOMO properties. In structural terms, this indicates that the LUMO locations are more noticeable and important than the HOMO locations. A molecule’s HOMO and LUMO energies are defined by the energy gap between them, which is abbreviated as ∆E. It’s an important metric that can reveal a lot about the molecule. Chemical reactivity is measured by the energy gap, which is the difference between the HOMO and LUMO energies. A molecule with a narrower gap is more reactive; the seven derivatives’ reactivity is: Altretamine(6.3764eV)\(\:>\)Flurabine(5.48690eV)\(\:\:>\)Cytarabine(5.42323eV)\(\:\:>\)Clofarabine(5.371eV)\(\:>\)Gemcitabine(5.31139eV)\(\:\:>\)Capcitabine(4.8381eV)\(\:>\)Dacarbazine (4.356 eV).

Fig. 1
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ESPM structures, DOS plots, optimized geometries and HOMO and LUMO structures.

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Fig. 2
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Optimized geometries, ESPM structures, DOS plots and HOMO and LUMO structures.

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Fig. 3
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Optimized geometries, ESPM structures, DOS plots and HOMO and LUMO structures.

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Fig. 4
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Optimized geometries, ESPM structures, DOS plots and HOMO and LUMO structures.

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Fig. 5
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Optimized geometries, ESPM structures, DOS plots and HOMO and LUMO structures.

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Fig. 6
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Optimized geometries, ESPM structures, DOS plots and HOMO and LUMO structures.

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Fig. 7
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Optimized geometries, ESPM structures, DOS plots and HOMO and LUMO structures.

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Due to its reduced energy gap, Dacarbazine has more reactivity than other oral chemotherapy drugs. Dacarbazine’s enhanced reactivity facilitates its involvement in chemical processes, potentially increasing its efficacy and susceptibility to interactions with other drugs. Conversely, Alteramine, with a greater energy gap, exhibits enhanced stability. The increased stability indicates that Alteramine is less prone to react with other substances, rendering it more dependable and constant in its physiological activity. Fundamentally, Dacarbazine has greater chemical reactivity, whereas Alteramine demonstrates enhanced chemical stability. An additional indicator of the regions vulnerable to electrophilic and nucleophilic attack is the two-dimensional distribution of HOMO and LUMO orbitals. The thermodynamical properties shown in Table 2, are extracted from Chemspider, PubChem and DFT.

Curvilinear regression modeling

Several topological indices are used to forecast chemotherapy medicines. Table 3 shows the values of the calculated topological indices. The QSPR approach examines regression models that are cubic, linear, and quadratic. The models are assessed in relation to six distance-based topological metrics. A correlation coefficient (R) between these indices and certain thermodynamical parameters as established by linear, quadratic, and cubic regression models is shown in Table 4. The regression model with the highest R-value is regarded as the most accurate when the correlation coefficient for a thermodynamical parameter is determined. Each of these tables shows the greatest value (R) for a different thermodynamical feature determined by the data analysis: linear, quadratic, and cubic. The values below 0.64 have been removed from the Tables out of convenience.

Table 2 Thermodynamical properties of chemotherapy drugs.
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Table 3 Distance-based topological indices of chemotherapy drugs.
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Table 4 Linear, quadratic and cubic correlation coefficient (R).
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Table 5 shows the optimal topological index for evaluating thermodynamical attributes with linear regression models. Figure 5 illustrates a graphic to this.

Table 5 Best estimation of linear regression models.
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The best topological index for evaluating thermodynamical attributes using quadratic regression models is shown in Table 6. Figure 9 shows a graphic that illustrates this.

Fig. 8
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Plots of best estimated linear regression models of thermodynamical properties with respect to topological indices.

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Table 6 Best estimation of quadratic regression models.
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The most efficient topological index for thermodynamical property estimation with cubic regression models is shown in Table 7. Figure 10 illustrates this with a diagram.

Fig. 9
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Plots of best estimated quadratic regression models of thermodynamical properties concerning topological indices.

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Table 7 Best estimation of cubic regression models.
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Using linear, quadratic, and cubic curvilinear models, the thermodynamic properties of chemotherapeutic drugs and the associated distance-based topological indices were investigated. Finding the correlation coefficient that most accurately reflected the attributes under investigation was the main purpose. The correlation coefficient R for the six distance topological indices based on the cubic, linear, and quadratic curvilinear models is displayed in Table 4. According to the specified topological indices, the best thermodynamical property predictions are shown in Tables 5 and 6, and 7 using linear, quadratic, and cubic regression models. For the regression models, you may have noticed that we bolded the correlation coefficient. The best quadratic and linear models of distance-based topological indices and their attributes are shown in Figs. 8 and 9, and 10, respectively. According to the findings, the association coefficients varied among the various topological indices and thermodynamical characteristics. When two variables are positively correlated, it means they have a strong tendency to move in the same direction. When they are negatively correlated, it means the reverse is true. A linear regression model’s independent variables are considered to have adequately explained the variance in the dependent variable when R2, also known as the coefficient of determination, is calculated. A perfect fit is indicated by a value of 1, varying from 0 to 1. Root mean squared error (RMSE) quantifies the degree to which the values predicted by the regression model correspond to the actual values. The properties which have the highest regression concerning topological indices are highlighted in bold. In Table 4, Every property has a corresponding topological index which predicts is best. In Tables 5, 6 and 7 we will conclude the best regression prediction according to the least RMSE value. Table 5 indicates that Heat Capacity shows the highest R2 = 0.979 with the least RMSE = 1.955, concerning the Gutman index with linear regression model: \(\:CV=.007Gut\left(G\right)+43.628\). Dipole moment and Sum of electronic and zero point energies are best predicted by multiplicative Harary index, with R2 = 0.658, 0.833 and least RMSE=1.340,113.2 respectively. Polarity, Zero-point vibrational energy, Enthalpy and Molar entropy are highly anticipated by the Wiener index with R2 = 0.779, 0.710, 0.734,0.972 and RMSE=2.026, 20.23,20.37, 20.37. Complexity is best predicted with the Harrary index with R2 = 0.979 and RMSE=63.35. Table 6 shows that Heat Capacity shows the highest R2 = 0.990 with the least RMSE=1.363, concerning the Wiener index with quadratic regression model:

Fig. 10
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Plots of best-estimated cubic regression models of thermodynamical properties concerning topological indices.

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.The Gutman index best predicted the dipole moment and Sum of electronic and zero point energies, with R2 = 0.693, 0.853 and least RMSE = 1.268,106.1 respectively. Polarity, Zero-point vibrational energy, Enthalpy, Molar entropy and Complexity are highly anticipated by the Wiener index with R2 = 0.783, 0.768, 0.786, 0.976, 0.781 and RMSE = 2.011, 18.07, 18.24, 2.847, 61.25 respectively. Table 7 shows that Heat Capacity shows the highest R2 = 1 with the least RMSE = 0.203, concerning the Wiener index with cubic regression model:

. Dipole moment, Polarity, Zero-point vibrational energy and Enthalpy are highly predicted by the multiplicated Harary index with R2 = 0.740, 0.918, 0.985, 0.988 and RMSE = 1.167, 1.236, 4.604, 4.339. The Harary index with R2 = 0.984 and RMSE = 2.337 best predicts molar entropy. Complexity is highly estimated by the Gutman index with R2 = 0.883 and RMSE = 44.82. Sum of electronic and zero point energies (SEZpE) and heat capacity are best estimated by the Schultz index with R2 = 0.860, 1 and RMSE = 103.6, 0.296 respectively. The sum of electronic and zero point energies (SEZpE) is also best predicted with the Wiener index with R2 = 0.860 and RMSE = 103.6. From all the above results it is concluded that the Wiener index is the best predictor and Heat Capacity is the best predicted property.

Model validation and evaluation

Heat capacity is considered the most accurately predicted property in QSPR modelling of chemotherapy drugs. Therefore, its actual values are compared with those predicted by linear, quadratic, and cubic models in Table 8, all of which show near-identical results with minimal errors. Table 9 depicts actual and predicted values of molar entropy. Both tables emphasize the accuracy of quadratic regression model.

Table 8 Actual and predicted values for heat capacity (CV).
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Table 9 Actual and predicted values for molar entropy(S).
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Conclusion

Our thorough investigation has shown that curvilinear regression models greatly improve the study of chemo drug action using molecular descriptors. We found that these models outperform linear regression models in terms of predictive power, particularly in cases when the underlying data shows nonlinear correlations. The models’ accuracy and resilience have been significantly enhanced by adding molecular descriptors as independent variables. The area of drug research and development stands to benefit greatly from our results. One way to cut down on development time and expenses is to utilize curvilinear regression models with molecular descriptors; this will help find and optimize medications that are more effective and selective. Additionally, unlike usual linear regression methods, our research highlights the significance of taking nonlinear connections between molecular descriptors and pharmacological action into account. Finally, future research might reveal how additional kinds of drugs can benefit from using curvilinear regression models and topological descriptors to forecast drug activity. To summarize, our research shows that curvilinear regression models are a strong tool for pharmacological activity analysis, especially when combined with molecular descriptors. The above findings are likely to be valid in bringing forth the potential improvement of chemo drug studies among curvilinear models. Though the study had acknowledged some limitations earlier, putting more weight on the potential for these models seemed appropriate- at least strong results were promised, rather than definitive claims. A more cautious conclusion probably would have underplayed potential impact, but results of this study may demand some optimism about curvilinear models, with future validation acknowledged. Our findings provide light on the molecular processes controlling drug action and lay the groundwork for better drug discovery pipelines.