Predictive modeling and regression analysis of diverse sulfonamide compounds employed in cancer therapy

Topological indices (TIs) have rich applications in various biological contexts, particularly in therapeutic strategies for cancer. Predicting the performance of compounds in the treatment of cancer is one such application, wherein TIs offer insights into the molecular structures and related properties of compounds. By examining, various compounds exhibit different degree-based TIs, analysts can pinpoint the treatments that are most efficient for specific types of cancer. This paper specifically delves into the topological indices (TIs) implementations in forecasting the biological and physical attributes of innovative compounds utilized in addressing cancer through therapeutic interventions. The analysis being conducted to derivatives of sulfonamides, namely, 4-[(2,4-dichlorophenylsulfonamido)methyl]cyclohexanecarboxylic acid (1), ethyl 4-[(naphthalene-2-sulfonamido)methyl]cyclohexanecarboxylate (2), ethyl 4-[(2,5-dichlorophenylsulfonamido)methyl]cyclohexanecarboxylate (3), 4-[(naphthalene-2-sulfonamido)methyl]cyclohexane-1-carboxylic acid (4) and (2S)-3-methyl-2-(naphthalene-1-sulfonamido)-butanoic acid (5), is performed by utilizing edge partitioning for the computation of degree-based graph descriptors. Subsequently, a linear regression-based model is established to forecast characteristics, like, melting point and formula weight in a quantitative structure-property relationship. The outcomes emphasize the effectiveness or capability of topological indices as a valuable asset for inventing and creating of compounds within the realm of cancer therapy.

The calculation of topological indices (TIs) for the mentioned compounds includes using their chemical configurations as well as depicting their molecular compositions.The notion of topological indices (TIs), pioneered by H. Wiener in 1947(Wiener, 1947), proves to be a valuable tool for characterizing the constructions of molecular graphs (Ramírez Alfaro, 2022).These indices provide quantitative measures that help describe the connectivity patterns within the compounds, offering insights into the structural features of the molecules.This information is crucial for understanding the relationships between molecular structures and properties, particularly in the context of anti-cancer compounds.Some of the topological indices we have discussed include first and second Zagreb (M 1 , M 2 and m M 2 ), harmonic (H), hyper Zagreb (HM), forgotten (F), reciporcal Randic (RR), Randic (RA), sum connectivity (S), geometric arithmetic (GA) and atom bond connectivity (ABC) index.
Numerical values, obtained from the molecular formulas of the compounds (Mohammed et al., 2016), are provided by these topological indices.These indices utilize mathematical algorithms based on the structural information encoded in the molecular graphs, offering quantitative insights into the compounds' topological features and connectivity patterns.Some interesting results on topological indices are characterized in (Natarajan et al., 2022;Zaman and He, 2022;Ullah et al., 2023a;Ullah et al., 2023b;Yan et al., 2023;Zaman et al., 2023;Arockiaraj et al., 2024a;Hayat et al., 2024a;Arockiaraj et al., 2024b;Hayat et al., 2024b;Arockiaraj et al., 2024c;Chidambaram et al., 2024).This numerical representation facilitates the characterization and comparison of different molecular structures, contributing to the understanding of their properties and potential activities, including anti-cancer effects.
Molecular descriptors find applications in diverse fields, including biology and mathematics (Aslam et al., 2017;Gutman et al., 2018).In this study, linear regression (Hosamani et al., 2017) employed to calculate various properties of these compounds, such as melting point (MP) and formula weight (FW), aiming to establish correlations between topological indices (TIs) and physicochemical characterizations.Linear regression analysis may be extended to higher-order regression models by using higher-order predictor variables.In basic linear regression, a linear equation represents the connection between predictors and response variables.To represent nonlinear connections, higher-order regression models include words like squared, cubed, and interaction.Leveraging the derived correlations, QSPR modeling (Duchowicz et al., 2008) will be conducted to accurately estimate the physical and chemical characteristics of the compounds (Hansen and Jurs, 1988).The significance of employing degree-based indices for QSPR analysis stems from their simplicity, resilience, interpretability, computational efficiency, adaptability, and compatibility with graph-based approaches.These indices give useful insights into the structural aspects of molecules and can help to construct predictive models for a variety of chemical attributes.
Numerous uses exist for topological indices (TIs) in the development of compounds.By examining the molecular structures graphs utilize these indices, analysts can point-out the most efficacious compounds (Shanmukha et al., 2022) for particular types of cancer and anticipate the toxicity and potential side effects of the compounds and contribute to the invention and creation of novel compounds.In summary, the using of topological indices (TIs) in compound exploitation has the capability to significantly advance the treatment of cancer and deepen our comprehension of molecular structures.
Compounds' molecular structures are depicted as graphs, with atoms as vertices and the connecting bonds as edges.The graph G(V, E) is a straightforward, limited, and related depiction of the composition of the compound, with V represents the collections of vertices and E as edges, respectively.In graphical theory, the degree of node in a graph, represented by "du," signifies the count of nodes adjacent to it.In chemistry, the valence of a compound corresponds to the degree of its associated node in the graph.The presented table (Table 1) includes information about topological indices, their notations, formulas, and the years in which they were introduced.

Results and discussions
Molecular descriptors find extensive applications in medicine, particularly in the domains of inventing and creating compounds.In the context of cancer treatment, the application of topological indices (TIs) becomes crucial for identifying potential compound candidates possessing the targeted physicochemical properties.Applying topologyrelated degree indices to compounds for the treatment of cancer allows for a deeper understanding of their structural characteristics and facilitates the correlation of these features with their biological activity.This approach provides valuable insights for the targeted design and discovery of compounds tailored for effective blood cancer treatment.
The quantitative structure-property relationship (QSPR) modeling approach proves beneficial for analyzing the relationship among molecular attributes and the physicochemical characteristics of compounds for addressing cancer.These aspects are instrumental in estimating the physical and chemical characteristics of newly discovered compounds, candidates according to their structural attributes, thus enhancing the efficiency of compound discovered.By leveraging QSPR modeling, researchers can gain valuable insights into how specific structural elements influence the properties of compounds, facilitating a more informed and targeted approach to identifying promising candidates for cancer treatment.
In this study, various compounds employed in cancer treatment underwent analysis utilizing topological indices and QSPR modeling.
The findings from the current analysis hold significant implications for advancing the creation of novel compounds in the treatment of cancer.Identifying the structural attributes and physicochemical properties of effective compounds provides valuable insights for designing new compounds with comparable attributes and potentially enhanced efficacy.
Moreover, this approach supports the enhancement of existing compounds through strategic modifications to their structural features, aimed at improving physical and chemical properties and augmenting its efficacy in the treatment of cancer.Analytical regression played a crucial role in the calculations conducted in this study.

Model of regression
The model of regression serves as a valuable tool in establishing relation between molecular attributes and the physico-chemical properties of compounds employed for addressing cancer.The results indicate a robust relationship between topological indices (TIs), the physical and chemical properties of these compounds.
The utilization of topological indices (TIs) in the context of cancer research is multifaceted.These descriptors prove instrumental in analyzing the structures of various compounds used in cancer treatment, spanning chemotherapeutic agents, targeted therapies, hormonal therapies, and immunotherapies.Analyzing topological indices in compound design aids in identifying new compounds and optimizing those already in existence.For instance, molecular descriptors enable the prediction of the efficacy of novel compounds in the treatment of cancer by scrutinizing their inherent structure.
Furthermore, Topological indices play a crucial role in determining the mechanism of action of compounds for addressing cancer, offering insightful understandings into the biological processes, underlying these compounds.In summary, the integration of topological indices (TIs) in cancer studies holds the potential to unearth new compounds and enhance the effectiveness of those already in use.The acquired outcomes undergo rigorous testing through mathematical expression 1.
Here, the sign P signifies a parameter associated with the physical and chemical properties of a compound.TI stands for some topological indices, whereas A, B represent the coefficients of regression utilized in this observation.With the aid of a linear QSPR model, the eleven TIs of potential cancer treatments are examined, as well as their physical characteristics.Using (1), we create a linear regression model for TIs of the potential compounds listed below.4dichlorophenylsulfonamido)methyl]cyclohexane carboxylic acid, then the following axioms holds;

Topological indices Notation Formula Introduced
First Zagreb Index M 1 (G) Gutman and Trinajstic' (Gutman and Trinajstić, 1972) M 2 (G) ( 1 du+dv ) Gutman, I. and Polansky, O. E. (Gutman and Polansky, 2012) ( 2 du+dv ) Graffiti (Fajtlowicz, 1988) Shirdel, H. Rezapour and A.M. Sayadi (Shirdel et al., 2013) Forgotten index Furtulaand Gutman, 2015 (Furtula andGutman, 2015) Reciprocal Randic Index RR(G) In 2014, Gutman, I., Furtula, B., and Elphick, C. (Gutman et al., 2014) In 1975, Million Randic (Farahani, 2013) Zhou and Trinajstić (Farahani, 2013) Shegehall and Kanabur (Vukičević and Furtula, 2009) Ernesto Estrade, 1998 (Das et al., 2011) Frontiers in Chemistry frontiersin.orgProof.Let G 1 belongs to 4-[(2,4-dichlorophenylsulfonamido) methyl]cyclohexane carboxylic acid, with the edge set represented as E and E 1 (u, v) denoting the set of edges in G 1 that adds degrees vertices "u" and "v," the frequencies are provided as follows: |E Then. i) By applying the first Zagreb index (M 1 ) and the provided edge partitions E 1 (u, v), we obtain: ii) By applying new version first Zagreb index (M 2 ) and the provided edge partitions E 1 (u, v), we obtain: iii) By applying second Zagreb index ( m M 2 ) and the provided edge partitions E 1 (u, v), we obtain: iv) By applying harmonic index (H) and the provided edge partitions E 1 (u, v), we obtain: 11.18 v) By applying hyper Zagreb index (HM) and the provided edge partitions E 1 (u, v), we obtain: vi) By applying forgotten index (F) and the provided edge partitions E 1 (u, v), we obtain: vii) By applying reciprocal randic index (RR) and the provided edge partitions E 1 (u, v), we obtain: viii) By applying randic index (RA) and the provided edge partitions E 1 (u, v), we obtain: 12.01 ix) By applying sum connectivity index (S) and the provided edge partitions E 1 (u, v), we obtain: Frontiers in Chemistry frontiersin.org05 Danish et al. 10.3389/fchem.2024.1413850 x) By applying geometric arithmetic index (GA) and the provided edge partitions E 1 (u, v), we obtain: xi) By applying atom bond connectivity index (ABC) and the provided edge partitions E 1 (u, v), we obtain: Proof.Let G 2 belongs to ethyl 4-[(naphthalene-2-sulfonamido) methyl]cyclohexane carboxylate with the edge set represented as E 2 and E 2 (u, v) denoting the set of edges in G 2 that adds degrees vertices "u" and "v," the frequencies are provided as follows: Then. i) Applying first Zagreb index (M 1 ) and the provided edge partition E 2 (u, v), we obtain: ii) By applying new version first Zagreb index (M 2 ) and the provided edge partition E 2 (u, v), we obtain: iii) By applying second Zagreb index ( m M 2 ) and the provided edge partition E 2 (u, v), we obtain: iv) By applying harmonic index (H) and the provided edge partition E 2 (u, v), we obtain: 14.39 v) By applying hyper Zagreb index (HM) and the provided edge partition E 2 (u, v), we obtain: vi) By applying forgotten index (F) and the provided edge partition E 2 (u, v), we obtain: vii) By applying reciprocal randic index (RR) and the provided edge partition E 2 (u, v), we obtain: viii) By applying randic index (RA) and the provided edge partition E 2 (u, v), we obtain: 15.56 ix) By applying sum connectivity index (S) and the provided edge partition E 2 (u, v), we obtain: x) By applying geometric arithmetic index (GA) and the provided edge partition E 2 (u, v), we obtain: xi) By applying atom bond connectivity index (ABC) and the provided edge partition E 2 (u, v), we obtain: Theorem 3. Let G 3 denotes the ethyl 4-[(2, 5dichlorophenylsulfonamido)methyl]cyclohexane carboxylate, then the following axioms satisfied for G 3. i ix. S (G 3 ) = 14.63 x. GA (G 3 ) = 30.38xi.ABC(G 3 ) = 24.76 Proof.Let G 3 belongs to ethyl 4-[(2,5-dichlorophenylsulfonamido) methyl]cyclohexane carboxylate with the edge set represented as E 3 and E 3 (u,v) denoting the set of edges in G 3 that adds degrees vertices "u" and "v," the frequencies are provided as follows: (2,4) | = 1.Then.i) By applying first Zagreb index (M 1 ) and the provided edge partitions E 3 (u, v), we obtain: ii) By applying new version first Zagreb index (M 2 ) and the provided edge partitions E 3 (u, v), we obtain: iii) By applying second Zagreb index ( m M 2 ) and the provided edge partitions E 3 (u, v), we obtain: 6.54 iv) By applying harmonic index (H) and the provided edge partitions E 3 (u, v), we obtain: 13.09 v) By applying hyper Zagreb index (HM) and the provided edge partitions E 3 (u, v), we obtain: vi) By applying forgotten index (F) and the provided edge partitions E 3 (u, v), we obtain: 4 1 2 + 3 2 + 9 2 2 + 3 2 + 3 2 2 + 2 2 + 9 1 2 + 4 2   + 4 3 2 + 4 2 + 1 4 2 + 4 2 + 2 3 2 + 3 2 + 1 2 2 + 4 2 522 vii) By applying reciprocal randic index (RR) and the provided edge partitions E 3 (u, v), we obtain: viii) By applying randic index (RA) and the provided edge partitions E 3 (u, v), we obtain: ix) By applying sum connectivity index (S) and the provided edge partitions E 3 (u, v), we obtain: 14.63 x) By applying geometric arithmetic index (GA) and the provided edge partitions E 3 (u, v), we obtain: xi) By applying atom bond connectivity index (ABC) and the provided edge partitions E 3 (u, v), we obtain: Proof.Let G 4 belongs to 4-[(naphthalene-2-sulfonamido)methyl] cyclohexane-1-carboxylic acid with the edge set represented as E 4 and E 4 (u,v) denoting the set of edges in G 4 that adds degrees vertices "u" and "v," the frequencies are provided as follows: 3,4 | = 4. Then.i) By applying first Zagreb index (M 1 ) and the provided edge partitions E 4 (u, v), we obtain: ii) By applying new version first Zagreb index (M 2 ) and the provided edge partitions E 4 (u, v), we obtain: iii) By applying second Zagreb index ( m M 2 ) and the provided edge partitions E 4 (u, v), we obtain: + 2 1 3 + 3 6.33 iv) By applying harmonic index (H) and the provided edge partitions E 4 (u, v), we obtain: By applying hyper Zagreb index (HM) and the provided edge partitions E 4 (u, v), we obtain: vi) By applying forgotten index (F) and the provided edge partitions E 4 (u, v), we obtain: vii) By applying reciprocal randic index (RR) and the provided edge partitions E 4 (u, v), we obtain: viii) By applying randic index (RA) and the provided edge partitions E 4 (u, v), we obtain: ix) By applying sum connectivity index (S) and the provided edge partitions E 4 (u, v), we obtain: x) By applying geometric arithmetic index (GA) and the provided edge partitions E 4 (u, v), we obtain: xi) By applying atom bond connectivity index (ABC) and the provided edge partitions E 4 (u, v), we obtain: Theorem 5. Let G 5 denotes the (2S)-3-methyl-2-(naphthalene-1sulfonamido)-butanoic acid, then the following axioms satisfied for G 5. i) By applying first Zagreb index (M 1 ) and the provided edge partitions E 5 (u, v), we obtain: ii) By applying new version first Zagreb index (M 2 ) and the provided edge partitions E 5 (u, v), we obtain: iii) By applying second Zagreb index ( m M 2 ) and the provided edge partitions E 5 (u, v), we obtain: iv) By applying harmonic index (H) and the provided edge partitions E 5 (u, v), we obtain: By applying hyper Zagreb index (HM) and the provided edge partitions E 5 (u, v), we obtain: vi) By applying forgotten index (F) and the provided edge partitions E 5 (u, v), we obtain: vii) By applying reciprocal randic index (RR) and the provided edge partitions E 5 (u, v), we obtain: viii) By applying randic index (RA) and the provided edge partitions E 5 (u, v), we obtain: 13.98 ix) By applying sum connectivity index (S) and the provided edge partitions E 5 (u, v), we obtain: x) By applying geometric arithmetic index (GA) and the provided edge partitions E 5 (u, v), we obtain: xi) By applying atom bond connectivity index (ABC) and the provided edge partitions E 5 (u, v), we obtain: The topological indices for five sulfonamide derivatives can be derived using a technique similar to that employed in Theorem 1, 2, 3,4, and Theorem 5, albeit with distinct topological indices.In Table 2, we have computed values for these indices, along with a comprehensive list of values for all medicines.

Models of regression for first zagreb index M 1 (G)
Melting Point = 401.739979445015-1.45786228160329[M 1 (G)] Formula Weight = 327.945883864337+ 0.185578108941419 [M 1 (G)]  3 provides the physicochemical characteristics of cancer medications, while Table 1 displays the calculated TI values derived from their molecular structures.The association correlation coefficients between TIs and two physicochemical characteristics are enumerated in Table 4. Figure 3 depicts the correlation among the topological index (TIs) and the physical and chemical characteristics of compounds, including their corresponding correlation coefficients.

Calculation of statistical metrics/ parameters
In our study, quantitative structure-property relationship (QSPR) modeling is conducted to establish a correlation between the physical and chemical properties of cancer compounds as well as their determined topological indices (TIs) of degree-based, the model of regression incorporates topological features as the independent variable.In this model, "B" stands for the constant of model, "r" indicates the correlation coefficient, and "N" signifies the number of sample compounds.
The theoretical and experimental computations highlighted in the tables emphasize a significant correlation coefficient.This testing methodology proves useful for comparisons between various models and evaluating their comparative enhancements.It is noteworthy that, in the majority of cases, the p-value exceeds 0.05, and the value    Tables 5-15 present the statistical parameters, with the abbreviations PP used for physiochemical properties, MP for melting point, and FW for formula weight.

Standard error of approximation (SE) and resemblance
Standard error (S.E), as presented in below table (Table 16), serves as an indicator of the extent to which an analysis deviates from the approximated regression line.It also offers insights into the accuracy of predictions derived from the regression line.For additional comparisons, both practically and in theory determined predictions of the models, focusing on their physical and chemical characteristics, are included in Tables 17,18.

Conclusion
The utilization of topological indices (TIs) and statistical parameters in direct quantitative structure-property relationship (QSPR) models has revealed robust correlation coefficients across various physicochemical characteristics of medications employed for addressing cancer.The outcomes of this analysis provide beneficial perspectives for the therapeutic industry, offering guidance within creating novel treatments and establishing safety precautions for cancer therapies.The noteworthy influence of correlation coefficients between diverse topological indices (TIs) for these medications underscores the capable to predict the physical and chemical characteristics of recently identified anticancer sulfonamides compounds, especially for addressing specific cancer conditions.These results hold promise for analysts engaged in pharmaceutical research, providing a potent tool for compound discovery and development.
In particular, our analysis revealed the greatest correlation value of r = .911for forgotten [F(G)] index with melting point, signifying its relevance.Additionally, the harmonic [H(G)] index demonstrated a substantial correlation of r = .21with formula weight, further contributing to the understanding of these medications' characteristics.This work not only contributes to our understanding of medications for cancer treatment but also offers practical implications for advancing pharmaceutical research and development in this critical area.In the near future, we aim to calculate the resistance distance based topological indices for the certain drugs.

Publisher's note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers.Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.Frontiers in Chemistry frontiersin.org13 Danish et al. 10.3389/fchem.2024.1413850 vii.RR (G 1 ) = 64.17viii.RA (G 1 ) = 12.01 ix.S (G 1 ) = 12.22 x.GA (G 1 ) = 25.38 xi.ABC(G 1 ) = 19.82

FIGURE 2 Chemical
FIGURE 2 Chemical Structures of derivatives of Sulfonamides as given in (A)-(E).

FIGURE 3
FIGURE 3Physicochemical properties and TIs.

TABLE 1
The considered topological indices, notations and formulas.

TABLE 2
Molecular descriptors for the candidate compounds.

TABLE 3
The physical characteristics of compounds.

TABLE 5
The utilization of statistical parameters in QSPR model for M 1 (G).

TABLE 6
The utilization of statistical parameters in the QSPR model for M 2 (G).
TABLE 10 The utilization of statistical parameters in the QSPR model for F(G).

TABLE 11
The utilization of statistical parameters in the QSPR model for RR(G).TABLE15The utilization of statistical parameters in the QSPR model for ABC(G).
TABLE 17 Comparison of actual and computed values for melting point from regression models.TABLE 18 Comparison of actual and computed values for formula weight from regression models.