Chemical Application of Topological Indices in Infertility Treatment Drugs and QSPR Analysis

The main challenges faced by medical researchers while producing novel drugs are time commitment, amplified costs, creating a safety profile for the medications, reduced solubility, and a lack of experimental data. Chemical graph theory makes an important theoretical contribution to drug development and design by investigating the structural properties of molecules. To improve drug research and assess the effectiveness of treatments, topological indices aim to provide a mathematical representation of molecular structures. In this study, the author examined a number of recently used drugs, including tamoxifen, mesterolone, anastrozole, and letrozole which are used to treat infertility. We compute the topological descriptors with the limiting behaviors associated with these pharmaceutical drugs and offer degree-based topological parameters for them. We conducted a QSPR investigation on the prospective degree-based topological descriptors using quadratic, cubic, exponential, and logarithmic regression models.


Introduction
According to consensus among the majority of specialists, infertility is defned as the inability to conceive after actively trying for at least a year.Infertility is a reproductive system disorder that prevents an individual from getting pregnant.Children play a key role in enabling their parents to contribute to the survival of the family, the culture, and the community.Most communities, especially in emerging nations, are designed with the expectation that younger generations will care for and support older family members in the future [1].Te availability of infertility cures is a component of Millennium Development Goal number 5, and infertility has been acknowledged by the WHO as a public health issue [2].Worldwide, approximately eighty million people sufer from impaired fertility, with prevalence rates ranging from less than fve percent to over thirty percent.By 2025, 7.7 million more people are anticipated to experience infertility, despite advancements in reproductive technologies [3].Male infertility is most frequently caused by abnormalities in the sperm shape, motility, or quantity (low sperm count) of sperm.Te majority of cases of infertility in people with ovaries are brought on by ovulation abnormalities, where ovulation is the process during which ovary releases an egg, getting it ready for potential fertilization by sperm.Approximately, 10-15% of couples in the reproductive age group face infertility, which difers from other health-related conditions as it also involves psychological and social factors.Approximately, 40-45% of cases of infertility are caused by men, either alone or in conjunction with their female spouses [4].Te rising interest in reproductive treatments has led to increased awareness and motivation for research regarding the psychological efects of infertility.Te connection between mental illness and infertility has been taken into account.In addition, studies have examined the psychological impacts of infertility itself along with chronic exposure to intrusive infertility therapies on mental health.Developing new pharmaceuticals is costly, time-consuming, and perplexing in this area.In addition, it requires quick diagnosis and treatment to address the condition.Among the ten selected medications that are essential for the community's health, safe, and highly efective, the medications include mesterolone, anastrozole, letrozole, tamoxifen, clomifene, progesterone, cabergoline, bromocriptine, goserelin, and estradiol.Figure 1 depicts the aforementioned medications.
TIs (topological indices) are distinct numerical descriptors derived from graphs that represent a chemical structure perfectly.Tese are efectively used to describe the physical characteristics of several medications.To achieve this, a variety of TIs and polynomials are used, and they accurately represent the veiled facts in graph theory.Every TI is signifcant and exhibits a notable function in chemical graphs.Te use of graph invariants (TIs) in investigations of quantitative structure-property relationships (QSPR) and quantitative structure-activity relationships (QSAR) has attracted signifcant attention in recent years.Topological indices are employed frequently in various domains such as mathematics, bioinformatics, and cheminformatics, but their most extensive utilization is observed in the feld of QSPR.Te optimal association between TIs and pharmacological characteristics can be determined using QSPR models.
Te author has determined degree-based TIs for infertility medicines in the current research paper.Similarly, topological indices and imposed QSPR (models) are used for comprehensive analysis.Tese TIs and drug characteristics have been estimated using a linear regression model (LRM).Te estimation of certain of these medications' physicochemical properties, or TIs, is also employed in the creation of QSPR models.Curve ftting was used in the QSPR research, which found a strong connection between the features of anticancer drugs.Te comprehensive examination of skin cancer drugs is conducted using Khan and Nadeem's imposed QSPR modeling.Tey have demonstrated the importance of topological indices in understanding the physical characteristics of medications [5].To anticipate the physicochemical characteristics of medications to treat diabetes, Parveen et al. [6] used the QSPR model.Earlier research on potential drugs for the treatment of COVID-19 was covered by Colakoglu.Given that discovery is a costly and complex phenomenon, this technique works best for predicting it [7].Targeted analysis and painstakingly crafted topological indexes were used by Parveen et al. to evaluate RA medications [8].Nasir et al. demonstrate a strong association between the properties of the pharmaceuticals and TIs in their results on blood cancer treatments using QSPR modeling [9].Frequent studies have initiated a direct association between the chemical properties and molecular structures of chemical compounds and pharmaceuticals.Topological indices are used for detailed investigations in the modeling of cardiovascular medicines [10].Drugs for vitiligo caused by autoimmune illness were discussed in [11].
Zaman et al. [12] conducted a thorough investigation of the neighborhood version of the sudoku nanosheet topological indices.Te issues with traditional drug delivery, such as poor solubility, toxicity, and irregular drug release, can be solved with the use of new technologies based on nanomaterials or naphtalenic nanosheets.Ullah et al. [13].Degreebased TIs are used by Zaman et al. in [14] to establish a best-ft regression model using QSPR, and they came to the conclusion that the indices are useful in predicting the characteristics of blood cancer medications.For cerium oxide nanostructures, modifed versions of the Second Zagreb index and other indices are computed by Zaman et al. [15].One of the supramolecular dyes employed in Masson's trichrome stain, fuchsine acid, has a wide range of uses in histology.As an organic semiconductor, it has numerous other signifcant uses in electronic felds and photonic devices.
With the ultimate goal of shedding light on the efcacy of the computed molecular descriptors for QSAR and QSPR investigations carried out by Ullah et al. [16], closed equations are derived for some of its signifcant irregular molecular descriptors.Ullah et al.'s [17] investigation is made simple by the engendered formulas and mathematical verdicts that are obtained.A number of graphs with excellent application-graph perspectives were estimated by Zhou et al. in [18], paving the path for new and established results in this feld.However, employing distance-base Eigen values and sign-less Laplacian energy of graphs, Indulal et al. in [19] produced signifcant fndings.Tis was succinctly explained by Kirmani et al. in [20].Generalization of descriptors may decrease the quantity of molecular graph-based descriptors while simultaneously improving current fndings and offering a stronger link to multiple molecular features.COVID-19 is a worldwide issue that is being studied and treated with a number of medications.Zhong et al. explore the illness medications by imposing QSPR modes and using topological indices to aid in their investigation [21].Jovanović and Stanić additionally take into account the spectral distances bounded by a certain constant in [22].Te aforementioned works have inspired us to explore the current study of infertility drugs using topological indices.

Methods
Te molecular graph depicts a molecular structure made up of a set of atoms or vertices V(G), which are joined by a set of bonds or edges E(G).In a molecular graph, size of graph is n and order m refers to the total number of atoms, or vertices, and the total number of bonds, or edges, respectively.Graph theory and chemistry are often employed to address various chemical graph problems.Topological indices play a crucial role in QSPR analysis as well as in the domains of chemical graph theory and mathematical chemistry.Te TIs we used were as follows: Defnition 1.ABC index [23] and Randic index [24] of G are given under Defnition 2. Sum connectivity index [25] and GA index [26] of G are given under 2 International Journal of Analytical Chemistry International Journal of Analytical Chemistry 3 (2) Defnition 3. Harmonic index [27] and hyper Zagreb index [28] of G are given under ( Defnition 4. Forgotten index [29] is defned as follows: Mesterolone with a chemical formula C 20 H 32 O 2 is a steroid used for the treatment of low testosterone levels, and it exerts minimal impact on sperm counts and levels.Anastrozole, with the chemical formula C 17 H 19 N 5 , serves as a nonsteroidal inhibitor prescribed for adjuvant therapy in postmenopausal women.It is used to reduce circulating estrogen.Aromatase inhibitors, including anastrozole, have become the preferred endocrine medications for postmenopausal breast cancer treatment.Letrozole is prescribed for the treatment of postmenopausal women, with a chemical formula of C 17 H 11 N 5 .Tamoxifen, with the chemical formula C 26 H 29 NO is a selective estrogen receptor modulator used in these treatments either alone or as an adjuvant.Clomifene is a medication used to stimulate ovulation, and its chemical formula is C 26 H 28 ClNO.Clomifene (formerly clomifene) is an ovulatory stimulant that is taken orally and operates as a selective estrogen receptor modulator.Clomifene can cause multiple ovulations, thereby increasing the likelihood of having twins.Tere is a potential for an elevated risk of ovarian cancer and weight gain.Clomifene can interact with tissues that have estrogen receptors such as the hypothalamus, pituitary, ovary, endometrial, vagina, and cervix.Cabergoline is used to treat hyperprolactinemic disorders caused by various factors, and its chemical formula is C 32 H 40 BrN 5 O 5 .Goserelin is used to treat breast and prostate cancer by lowering pituitary gonadotropin output.Estradiol is used to treat vaginal atrophy in menopause breast cancer treatment and advanced androgen-dependent prostate cancer.Te author has applied TIs on drugs in this article and obtained the required results.All the formulas for infertility dugs can be found at Pubchem and Chemspider.

Topological Indices Calculation
All other infertility drugs calculations are done with same procedure and are given in Table 1.

Dataset and Quantitative Structure Analysis and Regression Models
Tis section explores the use of topological indices and regression models to investigate the relationship between computed topological indices and physicochemical parameters.Te author has tabulated computations of TIs and physical characteristics of molecular structures in Table 1, respectively.Regression models can be constructed using these values.Te dataset for the aforementioned molecular structures includes the physicochemical characteristics found on ChemSpider, and the curves are ftted using regression models.In light of this, the author has investigated exponential, logarithmic, cubic, and quadratic models.Te square of the correlation coefcient's (R 2 ), the Fratio test, and signifcance (sig) were taken into account in the regression model table.Te most accurate regression model is the one with the highest R 2 value.Te topological index regression models for the specifc physicochemical attribute have a few best predictors, which have been highlighted here.Te regression model is the best option for this investigation.Instead of ftting straight lines, various regression models have been used.Tis process is referred to as a curvilinear regression analysis.Te following equations for curve ftting were examined in this study: where Y is dependent variable, a is the regression model constant, X i (i � 1, 2, 3, . ..) are independent variables, and 4 International Journal of Analytical Chemistry b i (i � 1, 2, 3, . ..) are the coefcients for descriptor.Te SPSS and MATLAB are useful for determining the results.In Table 2, the regression models for the enthalpy of molecular structures are presented whereas Figure 2 presents a logarithmic regression model portraying the association between H(G) and enthalpy, and Table 3 displays the regression models developed for predicting the boiling points of various molecular structures.Te table includes coefcients, statistical metrics, and other relevant information essential for understanding the relationships between molecular features and boiling points.Te models for predicting molar refractivity of molecular structures are summarized in Table 4. Te table provides a comprehensive overview of the regression coefcients, their signifcance, and any additional parameters employed in the models.Figure 3 illustrates the logarithmic regression model depicting the relationship between GA(G) and refractivity.As can be seen in Table 5, the author has constructed regression models that combined the aforementioned topological indices with the physical characteristics of molecular structures.Here Table 5 outlines the regression analysis for assessing the complexity of molecular structures.Te logarithmic regression model in Figure 4 demonstrates the relationship between GA(G) and molecular complexity.Te regression models for predicting the polarity of molecular structures are detailed in Table 6 while Figure 5 provides    6, the logarithmic regression model illustrates the relationship between S(G) and molar volume.

. Conclusions
Te author examined correlation coefcients between the topological indices and other physical characteristics of drugs used to treat infertility, which demonstrate how well the aforementioned indices serve as predictors.In particular, the study presents a quantitative structure-property relationship (QSPR) analysis of molecular descriptors (TIs), which are tools used to predict the physical and chemical characteristics of drugs, especially in the context of pharmaceutical and medical applications.It is worth noting that molar refractivity and complexity are reliable indicators for these predictions.However, the estimation of polarity and polar surface area is less dependable.
(i) In a quadratic and cubic regression model, the molecular descriptor S(G) is most accurately predicted by factors such as polarity, molar volume, and complexity.(ii) Molecular descriptor ABC(G) is best predicted by refractivity when using a logarithmic regression model.(iii) On employing an exponential regression model, molar refractivity is the most efective predictor for the molecular descriptors GA(G) and HM(G).
Summing up, the topological indices and the structural properties of infertility medication compounds have great and strong connections.Te correlation coefcients observed in QSPR modeling are located in close proximity to 1. Te observed result shows that p value is below 0.05 and Ftest value exceeds 2.5.Tese conditions confrm the validity of authors' fndings.Both experimental and theoretical model results are highly consistent with one another.Te author evaluates the predictive power of the degree base TIs using the physicochemical features of these structures.Our understanding of chemistry, pharmacy science, and drug discovery will all be improved by the fndings of this study.Employing the study's fndings, information on a certain chemical molecule can be found without the need for experiments if it is synthesized from these.
Tis research sets the stage for future investigations to calculate TIs for newly developed medications, providing valuable insights into their chemical structures and properties.Such knowledge can be crucial for the development and optimization of pharmaceutical compounds.Tis also enhances the feld of topological analysis and provides researchers with extra resources for studying molecule properties by introducing other indices.
In future, these indices can be utilized to many conversions of graphs and probe into other chemical networks and diseases drugs.Te molecular structures can also be analyzed on the base of graph energies this will also yield valuable insight into drugs discovery in pharmaceutical industry [30].

Figure 2 :
Figure 2: Logarithmic regression model of H(G) with enthalpy.

Table 1 :
Topological indices of infertility treatment drugs.

Table 2 :
Regression models for enthalpy of molecular structures.

Table 3 :
Regression models for boiling point of molecular structures.
a visual representation of the logarithmic regression model depicting the relationship between RA(G) and polarity.In Figure