Pharmacology has rapidly evolved, resulting in the introduction of numerous groundbreaking drugs each year. However, ensuring accurate testing performance requires the availability of appropriate equipment, a good rapport, and sufficient resources. Previous studies have shown that a drug’s chemical properties are intricately linked to its molecular structure. Pharmacological and medical researchers often utilize topological indices to examine the molecules’ properties and understand their impact on experimental outcomes. Hence, the topological index computation method is a useful tool for developing countries, allowing them to gather medical and biological data on upcoming drugs without the need for laboratory tests see for example [1,2,3,4].

Fibrates are a type of medication that have been shown to lower high levels of bad cholesterol (also known as low-density lipoprotein or LDL), increase good cholesterol (also known as high-density lipoprotein or HDL), and decrease the amount of small dense LDL particles in the blood. They have been found to be effective in reducing the mortality and morbidity associated with cardiovascular disease (CVD) in individuals who are at risk for developing it. However, conducting laboratory studies to investigate the physicochemical properties of fibrates can be both expensive and time-consuming. To overcome this challenge, chemists can use topological indices to derive mathematical equations that provide valuable insights into the properties of fibrates. For more information on fibrates, please refer to sources [5] and [6].

Chemical graph theory is a field that integrates mathematical modeling of chemical phenomena with graph theory. It utilizes topological indices to establish a correlation between the properties of a chemical molecule and its structure [7]. These indices are also known as graph invariants or graph-based molecular descriptors, and they quantify the topological features of a molecule or molecules [8]. The application of quantitative structure-property/structure-activity relationships (QSPR/QSAR) models, which are commonly employed in this field, allows for the prediction of molecular properties using these topological indices. In 1947, Harold Wiener introduced the Wiener index, the first topological index, Paraffin’s physical properties were determined using it [9].

Topological indices, which are numerical values derived from the molecular graph of a chemical compound, have been extensively studied in the fields of quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) analyses. These indices encode the structural and topological information of molecules and have proven useful in predicting various physical, chemical, and biological properties [10,11,12,13,14]. The use of molecular graphs to represent unsaturated hydrocarbon structures provides a more intuitive and comprehensive understanding of the molecular characteristics and behavior of compounds [15,16,17,18,19,20]. In drug design, knowledge of molecular structure is essential in determining their potential therapeutic activity and overall effectiveness. In this study, we examine several vertex-degree based topological indices, including the first and second Zagreb indices, hyper-Zagreb index, sigma index, Inverse symmetric deviation index, Max-min rodeg index, Min-max rodeg index, Inverse sum deviation index, Atom-bond connectivity index, Randic index, and Albertson index [21,22,23,24,25,26,27,28,29,30,31]. Additionally, we investigate topological indices based on distance, such as Wiener index, Schultz index, Harary index, and Gutman index [32,33,34]. These indices are used to classify the molecular descriptors and analyze the efficacy of curvilinear regression models in predicting the activity of fibrates drugs.

Molecular descriptors have been widely used to evaluate the physicochemical and bioactive properties of chemical structures, and their inclusion in curvilinear regression models can enhance the analysis of drug activity. Topological indices, such as the Zagreb indices, have shown promise in predicting the effectiveness of cancer treatments [35]. The max-min rodeg index has been found to give reliable predictions for octane isomers and polychlorobiphenyls in linear regression models [36]. A new index called the Atom-bond connectivity index has been proposed to determine the complexity of alkanes [37]. The first hyper-Zagreb index has been found to be the preferred method for estimating the boiling points of benzenoid hydrocarbons [38]. Additionally, the indeg indices have been applied to predict topological polar areas [39]. The inverse sum deviation index has been used to calculate the vaporization and sublimation enthalpies of monocarboxylic acids [40] and [41]. Irregularity indices based on different degrees, in addition to Albertson and Sigma indices, have been found to predict the physicochemical properties of octane isomers [42]. The Wiener index was first introduced in quantitative structure-property relationship (QSPR) studies, and has been shown to align well with the boiling points of alkanes [43]. The Wiener index has been further developed and used to explain different chemical and physical properties of molecules, as well as their biological activity [44]. The Schultz index has also been investigated to predict the boiling points of alkyl alcohols, and thus their suitability for various applications [45]. As indicated in Table 1, these indices are expressed mathematically and are shown with mathematical expressions.

Fenofibrate is an important component of a healthy diet and medication regimen, as it is used to reduce blood cholesterol and triglyceride levels. By decreasing triglyceride levels in the bloodstream, the risk of pancreatitis (inflammation of the pancreas) can be mitigated. To date, only one paper [46] has explored the use of topological indices in analyzing one of the drugs in the Fibrates family. This study utilized $ve-$degree, $ev-$degree, and degree-based ($D-$based) approaches to compute the topological indices of fenofibrate’s chemical structure. With limited existing literature on Fibrates that incorporate topological indices, this paper represents a pioneering effort in the investigation of novel physicochemical properties of Fibrates using this technique. In this work, Fenofibrate $\left(C_{20}{H}_{21}Cl{O}_4\right)$, Ciprofibrate $\left(C_{13}{H}_{14}C{l}_2{O}_3\right)$, Bezafibrate $\left(C_{19}{H}_{20}ClN{O}_4\right)$, Clofibrate $\left(C_{12}{H}_{15}Cl{O}_3\right)$ drugs used in the treatment patients with high cholesterol are studied.

Fibrates drugs are a class of medications commonly used to treat dyslipidemia, a condition characterized by abnormal lipid levels in the blood. Despite their widespread use, the molecular mechanisms underlying the activity of fibrates drugs are not well understood. One approach to addressing this challenge is to develop quantitative structure-activity relationship (QSAR) models that can predict the activity of fibrates drugs based on their molecular descriptors. In this study, we investigate the efficacy of curvilinear regression models in enhancing the analysis of fibrates drug activity through molecular descriptors. Curvilinear regression models are a type of non-linear regression model that can capture non-linear relationships between variables, making them useful for analyzing complex systems such as the interactions between drugs and their molecular targets. Our study builds upon previous research that has investigated the use of QSAR models to predict the activity of drugs. Several articles published in Symmetry have explored the use of topological indices and other mathematical methods to predict various properties of organic compounds, including their biological activity. For example, a study by Liu et al. [47] investigated the efficacy of using topological indices in QSPR models for predicting the densities and viscosities of biodiesel. The authors used a dataset of 105 biodiesel compounds with known properties and developed models using multiple linear regression and artificial neural network methods. They compared the performance of their models with previous studies and found that the models developed using topological indices had higher accuracy in predicting the properties of biodiesel. Zuo and Hu [48] developed QSPR models for predicting the melting points of organic compounds using molecular topology and quantum chemical descriptors. The authors used a dataset of 893 organic compounds and developed multiple linear regression models using the partial least squares (PLS) method. They compared their models with other models reported in the literature and found that their models were more accurate in predicting the melting points of organic compounds. Zhang et al. [49] developed QSPR models for predicting the melting points of organic compounds based on molecular topology. The authors used a dataset of 1,427 organic compounds and developed models using the neural network algorithm. They compared their models with other models reported in the literature and found that their models were more accurate in predicting the melting points of organic compounds. Naghipour and Kiasat [50] developed a QSPR model for predicting the fullerene-like behavior of C60 derivatives using topological indices. The authors used a dataset of 46 C60 derivatives with known fullerene-like behavior and developed a model using multiple linear regression. They compared their model with other models reported in the literature and found that their model had higher accuracy in predicting the fullerene-like behavior of C60 derivatives. Wang and Xu [51] developed QSPR models for predicting the boiling points of alkyl alkanes based on the novel vertex degree valence topological index. The authors used a dataset of 388 alkyl alkanes and developed models using multiple linear regression and artificial neural network methods. They compared their models with other models reported in the literature and found that their models were more accurate in predicting the boiling points of alkyl alkanes.

In our study, we apply curvilinear regression models to analyze the activity of fibrates drugs based on their molecular descriptors. By incorporating non-linear relationships between variables, we aim to enhance the accuracy and predictive power of QSAR models for analyzing the activity of fibrates drugs. Ultimately, our research may contribute to a better understanding of the molecular mechanisms underlying the activity of fibrates drugs, and to the development of more effective treatments for dyslipidemia. These studies demonstrate the usefulness of QSAR modeling and related techniques for predicting the activity of various compounds based on their molecular descriptors. By building on this previous work, we hope to further advance our understanding of the molecular mechanisms underlying the activity of fibrates drugs.

The QSPR model is a highly effective tool for predicting a wide range of physicochemical properties of drugs. To make these predictions, the model employs degree-based indices and distance-based topological indices (as detailed in Table 1). The properties considered include Polarizability, Sum of electronic and zero-point Energies, Sum of electronic and thermal Energies, Sum of electronic and thermal Enthalpies, Sum of electronic and thermal Free Energies, Zero-point vibrational energy, Complexity, Topological polar area, Dipole moment, Heat capacity, Molar entropy, and Octanol-water partition coefficients. To analyze the relationships between these properties and the topological indices, curvilinear regression (linear, quadratic, and cubic) is utilized. The model generates statistical parameters using SPSS and MATLAB statistical functions. In addition, DFT calculations are conducted at the B3LYP/6-31G(d,p) to gain insight into the optimized geometries, DOS plots, HOMO and LUMO orbitals energies, and distribution of the four derivatives studied in the next section. Section 3 examines the contributions of different topological indices as molecular structural descriptors. Finally, Section 4 concludes the paper.

In Figure 1,Figure 2,Figure 3, and Figure 4 four important characteristics of the four investigated Fibrate derivatives were indicated, included: $\left(1\right)$ optimized geometries, $\left(2\right)$ electron density mapped with electrostatic potential (ESPM), $\left(3\right)$ total density of states (DOS) plots, and $\left(4\right)$ the special distributions of the highest occupied molecular orbitals (HOMOs) and the lowest unoccupied molecular orbitals (LUMOs). Density Functional Theory (DFT) calculations of the investigated Fibrate derivatives utilized the one of the well-known hybrid functionals, Becke, $3-$parameter, Lee–Yang–Parr ($B3LYP$). In DFT, hybrid functionals incorporate a portion of Hartree-Fock exchange, as well as extra exchange from other sources (empirical/ab initio) to approximate the exchange-correlation energy. The B3LYP as a representation of Hamiltonian term in Schrödinger equation was combined with $6-31G(d,p)$ basis set as a representation of eigen-value wavefunction. It is a moderate double zeta ($\zeta$) basis set enlarged with two polarization basis functions, a $d-$function for heavy atoms (Carbon, Oxygen, and chlorine), and a p-function for all Hydrogen atoms. Most of the physicochemical properties of the investigated fibrate derivatives discussed in next section were obtained from the frequency calculations carried out at the same level of theory of optimization. Calculations were carried out using Gaussian 09 software suite [52]. Visualizations of molecular structures were performed by using GaussView (version $5.0.8$) [53], ESPMs were drawn used the Avogadro package [54], and GaussSum program [55] was used to DOS plots. ESPMs show how electron density is distributed in the four non-planar molecules considering the electrostatic potentials, and this gives information about the region in the molecule that has the highest or lowest electron density, and thus is most likely to be attacked by electrophilic or nucleophilic agents. Keep in mind that the nucleophilic and electrophilic attack regions are represented by blue (positively charged) and red colors (negatively charged). The red color is concentrated on the more electronegative atoms such as Oxygen (deep red) and chlorin atoms (light red), the blue color covered the Hydrogen atoms (the least electronegative atoms), while the Carbon atoms are covered by white color indicating intermediate electronegativity of Carbon atom. Thus, it is possible to determine the position and region in a molecule attacked by an electrophile or nucleophile using ESPMs. Molecule DOS plot indicates how many energy states electrons are allowed to occupy in the system. The HOMO energies of the four investigated Fibrate derivatives are $-6.230,-6.108,-6.166,$ and $-6.422$$eV$ for Fenofibrate, Ciprofibrate, Bezafibrate, and Clofibrate, respectively. Since, the HOMO energy used as a measure of electron-donating power of a molecule, destabilized HOMO (less negative) leads to more ability to donate electrons. The ability of electron donation of the five derivatives can be arranged as follows: Ciprofibrate > Bezafibrate > Fenofibrate > Clofibrate. On the other hands, the LUMO energy measures the ability of electron accepting of a molecule, more ability combined stabilized LUMO (more negative). Therefore, the derivatives ability to accept electrons is: Fenofibrate $(-1.720$$eV)>$ Bezafibrate $(-1.220$$eV)>$ Clofibrate $(-0.461$$eV)>$ Ciprofibrate $(-0.457$$eV)$. The energy gap (HOMO energy subtracted from LUMO energy) measures the chemical reactivity. Smaller gap is more reactive molecule, the reactivity of the four derivatives is: Fenofibrate $(4.51$$eV)<$ Bezafibrate $(4.95$$eV)<$ Ciprofibrate $(5.65$$eV)<$ Clofibrate $(5.96$$eV)$. Finally, the $2D-$special distribution of HOMO and LUMO orbitals is another indictor of the position/region subjected to electrophilic and nucleophilic attack. The HOMO and LUMO orbitals in Ciprofibrate are distributed on similar parts of molecule, except that the two chlorine atoms have more HOMO character. Other molecules, HOMO orbitals delocalized over different regions compared to the LUMO orbitals distribution.

In this section, the overall objective is to establish a quantitative structure-property-activity (QSPR) relationship between the various topological indices and some physicochemical properties/activity of the Fibrates drugs under study in order to assess the effectiveness of these drugs. Eleven degree-based and four distance topological indices were used for modeling antiviral activity. Based on DMol3-optimized geometries for Fibrates drugs investigated. The version $8.0$ of Material Studio from BIOVIA was used to perform DFT calculations, which are as follows: Polarizability $\left(P\right)$, Sum of electronic and zero-point Energies $\left(SE{Z}_{P}E\right)$, Sum of electronic and thermal Energies $\left(SETEnergy\right)$, Sum of electronic and thermal Enthalpies $\left(SETEnthalpy\right)$, Sum of electronic and thermal Free Energies $\left(SETFEnergy\right)$, Zero-point vibrational energy $\left(Z_{P}VE\right)$, Complexity $\left(C\right)$, Topological polar area $\left(TPA\right)$, Dipole moment $\left(DM\right)$, Heat capacity $\left(CV\right)$, Molar entropy $\left(S\right)$, and Octanol-water partition coefficients $\left(XlogP3\right)$ of several drugs currently being investigated for the treatment of high cholesterol which includes Fenofibrate, Ciprofibrate, Bezafibrate, Clofibrate drugs. It is possible to use curvilinear regression analysis to fit curves instead of straight lines, SPSS statistical software is used to analyze curvilinear regressions. As described below, the independent variables in the curvilinear regression models are topological indices. Indicators derived from cholesterol-lowering drugs. Based on the equations below, tests are conducted.

In this context, y represents the response or dependent variable, while a denotes the regression model constant, and $b_i\left(i=1,2,3\right)$ refers to the coefficients for each individual descriptor. The independent variable is represented by x, and n signifies the number of samples used in building the regression equation. $R^2$ denotes the coefficient of determination, R signifies the correlation coefficient, F represents the calculated value of the Fischer $F-$values test, $Se$ denotes the standard error of estimate, and $SF$ stands for $F-$significance. It should be noted that when the experimental and theoretical results are in close proximity to each other, the correlation coefficient approaches 1. To gauge the predictability of a model, it is necessary to compare the observed values and the model predictions, for which the Root Mean Square Error $\left(RMSE\right)$ metric is used. The predictive quality of a model is higher when the error or $RMSE$ is lower, which is calculated as follows: where $y_i$ is the observed value of the independent variable in the test set, ${\widehat{y}}_i$ is the predicted value of the independent variables in the test set, n is the number of samples in the test topological indices serve as independent variables. To evaluate our initial model, we used the $RMSE$ metric and then normalized the data to enhance our predictions’ accuracy. We measured the difference between predicted and actual values using the $RMSE$ score, which revealed that our model needed improvement. To address issues such as outliers and varying scales of measurement that could negatively affect model performance, we applied normalization techniques to our data. The normalization step was essential in improving the model’s accuracy, as it scaled variables to a common range, reduced the impact of outliers, and ensured that all variables were weighted equally. After normalization, we re-evaluated the model using the $RMSE$ metric, and the updated score showed a significant improvement in our predictions’ accuracy. Computed topological indices values are shown in Table 2. We compute the values using combinatorial computations and edge partitioning as follows: the molecular graph of Fenofibrate has 25 vertices and 26 edges. Its edges can be partitioned as $\left|E_{1,4}\right|=2,$$\left|E_{1,3}\right|=5,$$\left|E_{2,3}\right|=11,$$\left|E_{2,2}\right|=4,$$\left|E_{3,4}\right|=1,$$\left|E_{3,3}\right|=2,$ and $\left|E_{2,4}\right|=1.$ The molecular graph of Ciprofibrate has 18 vertices and 19 edges. Its edges can be partitioned as $\left|E_{2,2}\right|=2,$$\left|E_{1,4}\right|=4,$$\left|E_{2,4}\right|=2,$$\left|E_{1,3}\right|=2,$$\left|E_{2,3}\right|=6,$$\left|E_{1,4}\right|=2,$ and $\left|E_{1,3}\right|=1.$ The molecular graph of Bezafibrate has 25 vertices and 26 edges. Its edges can be partitioned as $\left|E_{1,3}\right|=4,$$\left|E_{2,3}\right|=11,$$\left|E_{2,2}\right|=6,$$\left|E_{3,3}\right|=1,$$\left|E_{2,4}\right|=1,$$\left|E_{3,4}\right|=1,$ and $\left|E_{1,4}\right|=2.$ The molecular graph of Fenofibrate has 16 vertices and 16 edges. Its edges can be partitioned as $\left|E_{1,3}\right|=2,$$\left|E_{2,3}\right|=6,$$\left|E_{1,2}\right|=1,$$\left|E_{2,2}\right|=3,$$\left|E_{3,4}\right|=1,$$\left|E_{1,4}\right|=2,$ and $\left|E_{2,4}\right|=1.$ Using MATLAB, it is possible to efficiently compute degree-based and distance-based topological indices, as explained in Algorithm 1 and Algorithm 2. To calculate the topological indices of molecules based on distance and degree, MATLAB utilizes various mathematical expressions. The Fibrates family and the drugs under consideration, namely Fenofibrate, Ciprofibrate, Bezafibrate, and Clofibrate, have been studied and are presented in Table 3, including their experimental data [52] and optimized geometries obtained through DFT calculations using the DMol3 module of Version 8.0 of Material Studio from BIOVIA. Table 4 shows the correlation coefficient $\left(R\right)$ between degree-based topological indices and some physicochemical properties, computed using a linear regression model. Quadratic regression model is used in Table 6 to calculate the correlation coefficient $\left(R\right)$ between these indices and some physicochemical properties. The cubic model is employed for this purpose in Table 8. Similarly, for the distance-based topological indices, linear, quadratic, and cubic regression models are utilized, and the results are presented in Table 10. Once the correlation coefficient for a physicochemical property is obtained, the model with the maximum R becomes the most accurate predictor of the regression model. This indicated in Table 5, Table 7, Table 9 and Table 11. By leveraging the power of MATLAB, it is possible to efficiently and accurately compute topological indices and use them to predict the physicochemical properties of molecules, which can be incredibly useful in various fields, including drug discovery and materials science.

Algorithm 1 Computational Procedure of calculation of degree-based indices

Input: Edges and nodes of molecule Output: $e\leftarrow$ Topological indices vector Step 1. Start Step 2. $G\leftarrow$ Graph of undirected edges Step 3. $A\leftarrow$ Adjacency matrix of G Step 4. $d\leftarrow$ Distances of G Step 5. $d_1\leftarrow$ Vertex degree of G Step 6. Calculate size of matrix d Step 4. Construct $A_N:$ for $i=1$ to number of columns do   for $j=1$ to number of rows do    if $i=j$ then      $A_N(i,j)=0$    elseif $A(i,j)=1$ then $A_N(i,j)=d_1\left(i\right)+d_1\left(j\right)$ First Zagerb index   $A_N(i,j)=d_1\left(i\right)\ast d_1\left(j\right)$ Second Zagerb index   $A_N(i,j)={(d_1\left(i\right)+d_1\left(j\right))}^2$ Hyper Zagerb index   $A_N(i,j)=\sqrt{{\displaystyle \frac{d_1\left(i\right)+d_1\left(j\right)-2}{(d_1\left(i\right)\ast d_1\left(j\right))}}}$ Atom Bond Connectivity index   $A_N(i,j)={\displaystyle \frac{1}{\sqrt{d_1\left(i\right)\ast d_1\left(j\right)}}}$ Randic index   $A_N(i,j)=\sqrt{{\displaystyle \frac{min(d_1\left(i\right),d_1\left(j\right))}{max(d_1\left(i\right),d_1\left(j\right)}}}$ min-max rodeg index   $A_N(i,j)=$$\sqrt{{\displaystyle \frac{max\left(d_1\left(i\right),d_1\left(j\right)\right)}{min(d_1\left(i\right),d_1\left(j\right))}}}$ max-min rodeg index $A_N(i,j)=|d_1\left(i\right)-d_1\left(j\right)|$ Alberston index $A_N(i,j)={(d_1\left(i\right)-d_1\left(j\right))}^2$ Sigma index $A_N(i,j)={\displaystyle \frac{d_1\left(i\right).d_1\left(j\right)}{d_1{\left(i\right)}^2+d_1{\left(j\right)}^2}}$ Inverse symmetric deg index $A_N(i,j)={\displaystyle \frac{d_1\left(i\right).d_1\left(j\right)}{d_1\left(i\right)+d_1\left(j\right)}}$ Inverse sum deg index    end if   end for  end for Step 5. $e=$ (summation of $A_N)/2.$