Ranking polycystic ovarian syndrome (PCOS) drugs using degree-based indices in QSPR models and CRITIC-driven MCDM methods

Abstract

Globally, 8–13% of reproductive-age women experience PCOS, a complicated endocrine condition. This study investigates several methods for treating polycystic ovary syndrome (PCOS) as well as its causes, which include a complicated interplay between hereditary susceptibility, hormonal imbalances, insulin resistance, and lifestyle variables. An advanced graph-theory technique is used in the investigation of PCOS medication chemical structure prediction, with special focus on unique degree-based topological indices like the Banhatti and Zagreb indices. The efficacy of fifteen drugs, such as metformin, letrozole, spirolactone, etc., is evaluated in this study using QSPR analysis. We rate these medications using three different multi-criteria decision-making (MCDM) algorithms: CRITIC, CoCoSo, and MABAC. According to the results of the CoCoSo and MABAC analyses, the medicine orlistat has the best chance of success, and the MCDM methods have improved the process of evaluating and ranking treatment choices. The correlation between indices and properties falls within the range of 0.8–0.9, which shows a strong positive correlation with the physicochemical characteristics of compounds. The study demonstrates how topological indexes might improve the process of finding novel drugs and creating individualised treatment regimens for PCOS patients.

Introduction

In the modern era, PCOS (polycystic ovarian syndrome) is a highly variable syndrome. One of the most common hormonal illnesses affecting women, it causes a variety of abnormalities in the female ovary. There are a number of tiny fluid-packed sacs inside the ovary. Clusters of these sacs resemble pearl-sized follicles and contain tiny cysts. An immature egg is present in every cyst¹. The size of these cysts ranges from 2 to 9 mm, which is considered minor². When 12 or more ovarian follicles are present, that type of PCOS is reported³. Approximately 60–70% of female PCOS patients have hirsutism, 60–80% of female PCOS patients have elevated testosterone levels, and 70–80% of female PCOS patients have metabolic abnormalities⁴. PCOS is one of the most common and unappreciated health conditions in the world today, affecting an estimated 8–13% of women of reproductive age. A normal ovary has regular menstrual periods and delivers one mature egg every cycle, but an ovary with PCOS has several little cysts and is unable to produce eggs on a regular basis. Moreover, a global comprehension of PCOS is vital as it is linked to enduring health hazards including metabolic syndrome, diabetes, cardiovascular issues, and mental wellness concerns. Affecting 8–13% of women in their reproductive years, polycystic ovary syndrome (PCOS) stands out as one of the most widespread endocrine conditions worldwide. The prevalence can be influenced by geographical location, diagnostic criteria, and ethnicity. For instance, research shows that PCOS is way more prevalent in “Middle Eastern and South Asian” communities, affecting up to 15–20% of women. The WHO partners with others to further the treatment and research of PCOS on a global scale.

Causes of PCOS, or polycystic ovarian syndrome, which is a complex disorder. Since hereditary traits increase the likelihood of hormone abnormalities, genetic predisposition is the primary contributor⁵. Stress is a big cause of PCOS since it increases cortisol levels, which disturbs hormonal balance. Insulin resistance is also a disorder in which the body’s incapacity to use insulin efficiently results in high insulin levels. An unhealthy lifestyle with poor eating and inactivity generates a vicious cycle that leads to weight increase and higher insulin resistance. The fundamental cause of PCOS, hormonal imbalance⁶, is linked to increased androgenic chemicals and altered reproductive hormones that produce irregular menstrual periods and issues with conception. As so, a comprehensive and customised PCOS management plan is required. Figure 2 shows PCOS (polycystic ovary syndrome) symptoms include mood problems connected to hormonal abnormalities, like anxiety and sadness⁷. Women with PCOS commonly suffer acne, particularly on the face, chest, and back, due to raised androgen levels that cause the skin to generate too much oil. Due to disturbed ovulation, unpredictable periods are prevalent, with cycles that might be erratic, excessively long, or even missing. A common cause of weight gain, especially around the abdomen, is insulin resistance⁸, which impairs the body’s ability to process glucose and raises the risk of type 2 diabetes. For many women with PCOS, infertility is a major issue because the hormonal imbalance makes it difficult to conceive by preventing normal ovulation. Elevated androgen levels can cause unusual hair growth (hirsutism), especially on the face, chest, and back. Sleep disorders such as sleep apnoea or insomnia, are also prevalent and frequently linked to weight gain and hormone problems. Finally, another typical symptom of PCOS is hair loss or thinning on the scalp, which is brought on by elevated androgen levels.

PCOS is frequently treated with a mix of medication and lifestyle⁹ modifications. Hormone balance and general health can be enhanced by taking vitamins, such as vitamin D and B vitamins. Anti-inflammatory treatments, such as dietary modifications and specific drugs, can help manage inflammation associated with PCOS. Removing toxins from the body by avoiding processed foods and chemicals helps reduce inflammation and assist the body’s natural detox processes. Since stress can exacerbate hormone abnormalities, it is crucial to reduce stress and anxiety. One might find benefit from methods including therapy or meditation. Managing insulin resistance requires management of blood sugar, which is often achieved with medication, a healthy diet, and lifestyle modifications. Apart from raising insulin sensitivity, consistent exercise can help regulate menstrual cycles and reduce the risk of diabetes acquisition. A well-rounded diet with whole foods, lean proteins, great fats, and lots of fruits and vegetables must be given top priority if one wants to properly treat PCOS symptoms, maximize insulin responsiveness, and increase general wellbeing. By including these strategies into an all-encompassing treatment plan, people with PCOS may regain control over their health and fitness, therefore significantly improving their condition. Managing insulin resistance requires blood sugar management, which is generally achieved with medication, a healthy diet, and lifestyle modifications. Apart from raising insulin sensitivity, consistent exercise can help regulate menstrual cycles and reduce the risk of diabetes acquisition.

Advancement of the chemical sciences depends on chemical graph theory, a branch of mathematical chemistry¹⁰. The concept of chemical graph theory is to understand the molecular composition of chemical compounds, so it may seem hard still. The fast expansion of the industries makes chemical analysis more challenging, which increases expenses and requires sophisticated techniques to determine the properties of a substance. As such, a fresh approach known as chemical graph theory was established¹¹. Using basic linked molecular graphs¹², chemical graph theory is a well-known subfield of graph theory inquiry that investigates the physicochemical and psychological properties of particular chemicals. Chemical graph theory explores chemical structures by use of topological indices is a numerical representation of molecular networks. Alan Randic and Harold Balaban created these indices in the middle of the twentieth century; they have evolved to incorporate molecular properties, including shape and connectivity. Important indices abound in degree-based, neighbourhood-based, eigenvalue-based, and distance-based forms. Advances in graph theory and computational tools have enabled more exact modelling of bioactivity and other chemical characteristics to be feasible, hence clarifying the link between structure and behaviour^13,14. Dr S.K. Banhatti first presented Banhatti indices in 2006. His studies focused on degree-based topological indices of chemical graph theory, which relate various chemical characteristics to molecule structure. Cheminformatics and computational chemistry have then regularly employed these signals to forecast molecular activity. Banhatti is the degree-based topological index language. Their definition in a molecular network derives from the vertex degree. Molecular attributes like boiling point, stability, and chemical reactivity are projected in QSAR/QSPR models using banhatti indices. To understand molecular complexity and how atom interactions affect the chemical behaviour of a molecule, one must give great thought to the relevance of indices. Their main responsibilities in disciplines such as cheminformatics, material science, and drug development are those of predicting molecular characteristics. Among the several various kinds of Banhatti indices are the K-hyper¹⁵ Banhatti index, the first Banhatti index, the second modified Banhatti index, and the first modified Banhatti index. By means of QSPR technology, researchers may predict certain features of a molecule based on its unique structure. Molecular descriptors are numerical representations of molecular properties linked to solubility, toxicity, or boiling point. QSPR relates structural information to desired physical or chemical characteristics, therefore allowing one to forecast new compound behaviour. Multi-criteria decision making (MCDM) or multi-criteria decision analysis (MCDA) is one of the most accurate ways of decision-making, and it might be known as an evolution in this sector^16,17. One of the first research studies on multi-criteria decision-making was established by Benjamin Franklin when he published his research on the moral algebra notion. Since the 1950s, a number of theoretical and empirical researchers have studied MCDM techniques to investigate their mathematical modelling potential in order to offer a framework that can assist in organising decision-making issues and producing preferences from options¹⁸. CRITIC, COCOSO AND MABAC are the widely used methods in the domain of multi-criteria decision-making.

This paragraph represents some advances in the literature in different fields. Using QSPR analysis, Hasani and Ghods’¹⁹ study investigates topological indices derived from M-polynomials to forecast the physicochemical characteristics of beta-blocker medications. Strong connections are seen in the study, with indicators such as the harmonic index and ReZG₃ doing exceptionally well in forecasting important characteristics like boiling point and polarisability. Kirmani et al.²⁰ research has assessed the physicochemical characteristics and biological activity of antiviral medications, such as remdesivir and chloroquine, for the treatment of COVID-19 using topological indices. Using QSPR models based on the ZEP topological index, the study examined the molar refraction, polarisability, and refractive index of symmetric and asymmetric monocarboxylic, dicarboxylic, and unsaturated monocarboxylic acids. The models were validated using y-randomisation tests, leave-one-out cross-validation, and external validation²¹. Since then, it has been applied to link molecular structure to biological activity and to explain a variety of molecular physical and chemical properties²². For the study, nine anti-malarial drugs’ degree-based topological indices were computed; subsequently, regression models were developed to connect the indices to the physicochemical properties of six of the drugs. The results showed that the regression models effectively linked the topological indices to the physicochemical parameters and that statistical analysis provided understanding of the relationships between chemical structure and characteristics. In drug design QSAR/QSPR studies, these findings enhanced the application of topological indices²³. In order to better grasp the link between molecular structure and characteristics in benzenoid systems, the work computed six significant degree-based topological indices for the molecular graphs of benzenoid hydrocarbons. It proved that the physical characteristics of the hydrocarbons strongly correlated these indicators²⁴. Topological indices offer a reasonably priced replacement for traditional laboratory testing for the aim of projecting the physicochemical properties of molecular molecules. These indices have been shown to effectively link certain chemical properties to molecular features, including bond length, strength, and shape¹⁰. For understanding the properties of dendrimer structures and chemical compounds, topological indices including the Banhatti and Zagreb indices are invaluable tools. Previous work has focused on constructing polynomials and studying their interactions to enhance property predictions and better understand molecular characteristics²⁵. This review highlighted the need for stronger theoretical foundations and more research on fuzzy MCDM methods, focusing on 72 studies on MCDM methods in COVID-19 research, with the most popular approaches being AHP (fuzzy AHP), TOPSIS, and VIKOR. Fuzzy logic was frequently used to manage uncertainty, despite the computational challenges it presented. India, Turkey, and China were the main contributors to this review²⁶.

Recent studies have explored fuzzy-based topological indices, such as the first Zagreb index on fuzzy graphs²⁷, the hyper-Zagreb index in a fuzzy environment²⁸, neighbourhood and competition graphs under fuzzy incidence²⁹, the hyper-Wiener index for fuzzy graphs³⁰, and an investigation of edge F-index on fuzzy graphs and application in molecular chemistry³¹. These indices have demonstrated their applicability in various fields, including molecular chemistry and financial analysis. However, this study focuses specifically on degree-based and Banhatti indices for QSPR analysis, rather than fuzzy indices. The integration of fuzzy approaches remains an interesting direction for future research.

Basic definitions

A graph (G) is a mathematical structure with the formula G = (V, E), where the set of edges (connections between vertices) can be expressed by E and the set of vertices (nodes) by V. Whereas the edges of a directed graph are ordered pairs e = (u, v), the edges of an undirected graph are unordered pairs e = {u, v}. The number of edges that are connected to a vertex is its degree (dr). It is divided into in-degree (edges entering the vertex) and out-degree (edges leaving the vertex) in directed graphs. Chemical bonds between atoms are represented by edges, and atoms are represented by vertices in chemical graph theory.

First Zagreb index

The sum of the degrees of neighbouring vertices for every edge in a graph is the definition of the degree-based first Zagreb index. The first Zagreb index was introduced by Ivan Gutman and Nenad Trinajstic³². In terms of mathematics, it is stated as:

$$M_{1} G = \mathop \sum \limits_{{\left\{ {u,v} \right\} \in E}} \left( {d\left( u \right) + d\left( v \right)} \right).$$

Second Zagreb index

The structure of molecules represented as graphs is described by the second Zagreb index, a topological index used in graph theory, especially chemical graph theory. It is also introduced by Ivan Gutman and Nenad Trinajstic in 1972³³. It is described in terms of the graph’s neighbouring vertices degrees. The second Zagreb index can be stated mathematically as follows:

$$M_{2} \left( G \right) = \mathop \sum \limits_{{\left\{ {u,v} \right\} \in E}} \left( {d\left( u \right) \times d\left( v \right)} \right).$$

Second modified Zagreb index

In graph theory, the second modified Zagreb measure is a degree-based topological measure. The modified second Zagreb index was introduced by Nikolic et al.³⁴. It is described as:

$$mM_{2} \left( G \right) = \mathop \sum \limits_{{\left\{ {u,v} \right\} \in E}} \frac{1}{du \times dv} .$$

Redefined third Zagreb index

The redefined third Zagreb index was proposed in 2013 by Ranjini et al.^35,36. This is the definition of the Redefined third Zagreb Index:

$$ReZG_{3} \left( G \right) = \mathop \sum \limits_{{\left\{ {u,v} \right\} \in E}} \left( {d\left( u \right) \times \left. {dv} \right)} \right) \times \left( {d\left( u \right) + d\left( v \right)} \right).$$

Forgotten index

The sum of the squares of the degrees of the vertices for each edge in the graph G is known as the forgotten topological index, or F(G). It was introduced by Furtula and Gutman³⁷. It is mathematically stated as:

$$F\left( G \right) = \mathop \sum \limits_{{\left\{ {u,v} \right\} \in E}} \left( {\left( {du} \right)^{2} + \left( {dv} \right)^{2} } \right).$$

Inverse sum index

The inverse sum index was introduced by Vukicevic, Furtula and Gasperov^38,39,40. It is mathematically stated as:

$$I\left( G \right) = \mathop \sum \limits_{{\left\{ {u,v} \right\} \in E}} \frac{du \times dv}{{du + dv}} .$$

Symmetric division index

The symmetric division was also introduced by Vukicevic, Furtula and Gasperov^38,39,40. The definition of symmetric division index is mathematically stated as:

$$SSD\left( G \right) = \mathop \sum \limits_{{\left\{ {u,v} \right\} \in E}} \frac{{\left( {du} \right)^{2} + \left( {dv} \right)^{2} }}{du \times dv}{ }.$$

Harmonic index

The Harmonic index was introduced by Fajtlowicz⁴¹. The harmonic index is defined as follows:

$$H\left( G \right) = \mathop \sum \limits_{{\left\{ {u,v} \right\} \in E}} \frac{2}{du + dv} .$$

K-Banhatti indices

The Banhatti indices were first presented in 2016 by Indian mathematician Kulli¹⁵. Milan Randic’s 1972 introduction of the Zagreb indices served as Kulli’s inspiration. The features of chemical and nonchemical graphs are correlated using Kulli’s Banhatti indices. He used these indices to investigate how connected graphs, including path, cycle, complete, and bipartite graphs, behave. These indices can be used to compare the various graph operations and to characterise the characteristics of path graphs⁴². These indices are used to investigate the structure of well-known Jahangir graphs and various nanomaterials. Banhatti indices and Revan indices are used to study a particular SiC structure, named SiC3–I [s, t]^43,44. The M-polynomial of the Jahangir graph has been used to calculate several significant topological indices, including the atom bond connectivity index, geometric arithmetic index, K-Banhatti indices, hyper K-Banhatti indices, and modified K-Banhatti indices⁴⁵. As descriptors, Banhatti indices are used to predict a molecule’s physicochemical characteristics, including stability, reactivity, and boiling temperatures. Suppose that $d_{G} \left( e \right)$ is the degree of the edge e in graph G. It is mathematically defined as:

$$d_{G} \left( e \right) = D_{G} \left( u \right) + D_{G} \left( v \right) - 2$$

First and second K-Banhatti index

K Banhatti index B1(G) is the sum of the degrees of the incident element pairs, and B2(G) is the sum of the products of the degrees of the incident element pairs of G. Mathematically they can be represented as:

$$\begin{aligned} B_{1} \left( G \right) & = \mathop \sum \limits_{uv \in \left( G \right)} \left( {d_{u} + de} \right), \\ & = \mathop \sum \limits_{uv \in \left( G \right)} \left[ {\left( {du + de} \right) + \left( {dv + de} \right)} \right]. \\ {\text{B}}_{{2{ }}} \left( {\text{G}} \right) & = \mathop \sum \limits_{uv \in \left( G \right)} \left( {d_{u} + d_{e} } \right), \\ & = \mathop \sum \limits_{uv \in \left( G \right)} \left[ {\left( {du \times de} \right) + \left( {dv \times de} \right)} \right]. \\ \end{aligned}$$

Modified first and second K-Banhatti index

The purpose of these modified indices was to improve the K-Banhatti indices analytical capabilities. Numerous studies have examined the structural characteristics of chemical compounds and other complex networks using these indices. The modified indices in molecular structures can give a more thorough understanding of the stability and reactivity of chemical compounds by accounting for characteristics such as electron densities, atomic masses, and bond energies. In terms of mathematics, they are expressed as:

$$\begin{aligned} mB_{1} \left( G \right) & = \mathop \sum \limits_{uv \in \left( G \right)} \frac{1}{{d_{u} + d_{e} }}, \\ & = \mathop \sum \limits_{uv \in \left( G \right)} \left[ {\frac{1}{{\left( {du + de} \right)}} + \frac{1}{{\left( {dv + de} \right)}}} \right]. \\ mB_{2} \left( G \right) & = \mathop \sum \limits_{uv \in \left( G \right)} \frac{1}{{d_{u} \times d_{e} }}, \\ & = \mathop \sum \limits_{uv \in \left( G \right)} \left[ {\frac{1}{{\left( {du \times de} \right)}} + \frac{1}{{\left( {dv \times de} \right)}}} \right]. \\ \end{aligned}$$

Methodology

Evaluation of the QSPR modelling results is done through regression analysis with the help of SPSS software. The main statistical technique for determining and measuring correlations between the independent variables (molecular descriptors) and the dependent variable (property of interest) is regression analysis. Besides regression analysis, we examine the outcomes using three widely used multi-criteria decision-making (MCDM) approaches. Microsoft Excel is used to handle the data for various MCDM techniques, and we determine scores and rankings for each alternative under each method. The results are presented in a clear and comprehensive way; the processed data is organised in tables and displayed using charts. This systematic approach provides an easily comprehensible comparison of the best rankings obtained from the different MCDM methodologies. These methods are MABAC, CoCoSo, and CRITIC. By combining MCDM approaches with SPSS regression analysis, our analysis ensures the correctness and dependability of the results and offers a thorough knowledge of QSPR modelling outcomes.

Materials

The Chemspider database helped to acquire the chemical structures of fifteen drugs for PCOS. PNG format shows 2D structures presented in Table 1 on the given website. In the analysis, we included the drugs Metformin (DB00331), Letrozole (DB00977), Ethinylestradiol (DB00977), Levonorgestrel (DB00573), Spironolactone (DB00421), Dexamethasone (DB00483), Pioglitazone (DB01238), Lorcaserin (DB08891), Finasteride (DB01050), Eflornithine (DB00216), Orlistat (DB00323), Tasimelteon (DB08935), Flutamide (DB00575), Rosiglitazone (DB00411) and Oxandrolone (DB00289) with their corresponding Blood Bank IDs to examine their effectiveness in PCOS. Table 2 displays the computed calculations of the degree-based topological and Banhatti indices by using the above formulas. Molecular structures shown in chemical diagrams are crucial for comprehending the physicochemical characteristics that affect the way pharmaceuticals behave in the body while designing medications for PCOS. By examining the molecular characteristics of possible treatment drugs, quantitative structure–property relationship (QSPR) modelling is essential for forecasting important elements including toxicity, absorption, distribution, metabolism, and excretion (ADME). Maximising the absorption of drugs depends on QSPR, which also guarantees that drugs reach intended tissues and reduces adverse effects. Forecasting drug interactions helps one to design drugs with improved effectiveness and safety aspects more easily. By designing more tailored PCOS therapies utilising QSPR analysis and enhancing pharmacological effectiveness, regulatory compliance, and general patient safety, researchers can save time and money. Based on physio-chemical characteristics including molar mass (MW), boiling point (BP), enthalpy of vaporisation (E), molar refractivity (MR), polarisability (P), molar volume (MV), and ACD/logP retrieved from ChemSpider, the QSPR model is displayed in Table 3.

Table 1 Molecular structures of polycystic ovarian syndrome drugs.

Table 2 Calculations of degree based topological and Banhatti indices (decision matrix).

Table 3 Physio-chemical properties of drugs.

Apply linear regression in QSPR analysis

The QSPR study of major topological descriptors that are strongly connected to the features of biological agents used to treat PCOS is the primary focus of this work. Through the use of linear regression, it provides insight into the structural aspects determining the therapeutic potential of PCOS drugs by exposing the complicated correlations between their physicochemical features and related topological indices. A chosen set of topological descriptors that were developed using degree-based calculations and reflected major structural properties of the molecular networks of PCOS medicines were among the independent variables in the linear regression modelling applied for this investigation. These descriptors, which detailed the compound’s molecular interactions, branching patterns, and overall structural complexity, supplied a mathematical framework for understanding how the chemicals influenced medication performance. The dependent variables were the physicochemical features of the PCOS medications, including their binding affinity, bioavailability, and solubility. Maintaining the effectiveness of medication needs these features. To carefully analyze and quantify the correlations between pharmacological features and molecular structure, a thorough linear regression model was created utilising the sophisticated analytical capabilities of SPSS software. This technique aided the discovery of important structural variables and enabled the building of a prediction framework for improved PCOS medication research and development. The following linear equation was used to compute the linear regression model’s equation.

$$P = q + t\left( {TI} \right)$$

(1)

In the equation, (P) represents the predicted property, such as the physio-chemical property of a compound, (q) is constant, (t) is the coefficient of regression analysis, (TI) is the topological indices. F statistics (F value), correlation coefficient (R), P statistics (P value), coefficient of determination ($R^{2}$) and $S_{res}$ (standardised residuals). In linear regression, a standardised residual is a residual split by its standard deviation, which is used to evaluate model fit and find outliers, and N is typically the number of data points are parameters used for the regression analysis. (P ≤ 0.05) indicates that the relationship between the predictors and response variable is significant, (F > 2.5) shows that the model fits data better than a model without any predictors, and ideal R should be close to 1 (for a strong positive relation) to − 1 (for a negative relation), and values above 0.8 indicate a strong correlation. The regression equations and statistical parameters for the degree-based topological indices are shown in Tables 4, 5, 6, 7, 8, 9, 10 and 11, and the Banhatti indices linear regression model are shown in Tables 12, 13, 14 and 15. Correlation coefficient (R) for properties of PCOS drugs is labelled in Table 16. All correlation R values for defined properties and indices were shown graphically in Figs. 1 and 2.

Table 4 Statistical calculation of topological index (M₁(G)).

Table 5 Statistical calculation of topological index (M₂(G))).

Table 6 Statistical calculation of topological index (mM₂(G)).

Table 7 Statistical calculation of topological index (ReZG₃(G)).

Table 8 Statistical calculation of topologic index (F(G)).

Table 9 Statistical calculation of topologicndex (I(G)).

Table 10 Statistical calculation of topologil index (H(G)).

Table 11 Statistical calculation of topological index (SDD(G)).

Table 12 Statistical calculation of Banhatti index (B₁(G)).

Table 13 Statistical calculation of Banhatti index (B₂(G)).

Table 14 Statistical calculation of Banhatti index (mB₁(G)).

Table 15 Statistical calculation of Banhatti index (mB₂(G)).

Table 16 Correlation coefficient R values for properties of PCOS drugs.

Fig. 1

Graphical depiction of correlation R-values among properties and indices.

Fig. 2

All correlation R values for various attributes and indices are compared.

Multi criteria decision making technique (MCDM)

Methodology of CRITIC method and implementation

The CRITIC (Criteria Importance Through Intercriteria Correlation) method is used in multi-criteria decision-making (MCDM) to objectively calculate the weights of criteria based on the correlation between criteria and the intensity of contrast. This strategy allows for a data-driven and objective approach, which is especially useful when the criteria weights are not pre-defined by experts or decision-makers. Abid Mehboob et al.⁴⁶, recently studied the MCDM technique by using CRITIC method for weight determining and CoCoSo and MABAC method for determining the rank of the brain tumor drugs. Here, the CRITIC approach forms the basis of an advanced framework for decision-making that aims to choose the best drugs for treating PCOS. Its use entails evaluating a number of factors, including patient suitability, cost-effectiveness, accessibility, adverse effect profile, and effectiveness of therapy. The methodology unbiasedly allocates weights that represent the relative significance of each criterion by examining these elements through the CRITIC lens, guaranteeing that the decision-making process stays grounded scientifically. These steps are applied using quantitative tools and statistical computations.

Step 1 Create the decision matrix shown in Table 2. Define a set of $m$ alternatives $\left( {A_{1}, A_{2}, \cdots , A_{m} } \right)$ and n criteria ($C_{1} ,C_{2} ,C_{3} , \ldots ,C_{n } )$. Populate the matrix V with values $V^{\prime}_{ij}$, where $V^{\prime}_{ij}$ is the performance of alternative $A_{i}$ with respect to criterion $C_{j}$.

$$V = \left[ {v_{ij} } \right]_{m \times n} = \left[ {\begin{array}{*{20}c} {v_{11} } & {\quad v_{12} } & {\quad \cdots } & {\quad v_{1n} } \\ {v_{21} } & {\quad v_{22} } & {\quad \cdots } & {\quad v_{2n} } \\ \vdots & {\quad \vdots } & {\quad \ddots } & {\quad \vdots } \\ {v_{m1} } & {\quad v_{m2} } & {\quad \cdots } & {\quad v_{mn} } \\ \end{array} } \right];\quad i = 1,2,3 \ldots ,m\;{\text{and}}\;j = 1,2,3 \ldots ,n$$

(2)

Step 2 Normalize the decision matrix. Normalization ensures all the criteria values are scaled to a comparable range (usually [0,1]). For each element $v^{\prime}ij$ for matrix:

• 1. For benefit criteria (higher is better):

  $$v^{\prime}ij = \frac{{vij - {\text{min}}\left( {vi} \right)}}{{\max \left( {vi} \right) - {\text{min}}\left( {vi} \right)}}.$$

  (3)

• 2. For cost criteria (lower is better):

  $$v^{\prime}ij = \frac{{\max \left( {vi} \right) - vij}}{{\max \left( {vi} \right) - {\text{min}}\left( {vi} \right)}}.$$

  (4)

  where $vij$ is the normalized value of the ith alternative across the jth attribute.

Step 3 The standard deviation is calculated by the following equation.

$$\sigma_{j} = \sqrt {\frac{1}{n - 1}\mathop \sum \limits_{j = 1}^{n} \left( {v^{\prime}_{ij} - \overline{{v^{\prime}_{i} }} } \right)^{2} } ;\quad i = 1,2,3, \ldots ,m$$

(5)

where $\overline{v^{\prime}j}$ is the mean of the normalized value of criterion $i$.

The criterion has greater discriminating power if the variability is higher.

Step 4 Determine the correlation between criteria.

Compute the correlation coefficient $d_{ik}$ between every pair of criteria j and k using:

$$d_{ik} = \frac{{\mathop \sum \nolimits_{j = 1}^{n} \left( {v^{\prime}_{ij} - \overline{{v^{\prime}_{j} }} } \right)\left( {v^{\prime}_{kj} - \overline{{v^{\prime}_{k} }} } \right)}}{{\sqrt {\mathop \sum \nolimits_{j = 1}^{n} \left( {v^{\prime}_{ij} - \overline{{v^{\prime}_{j} }} } \right)^{2} \mathop \sum \nolimits_{j = 1}^{n} \left( {v^{\prime}_{kj} - \overline{{v^{\prime}_{k} }} } \right)^{2} } }}.$$

(6)

$$\overline{{v^{\prime}_{j} }} = \frac{1}{n}\mathop \sum \limits_{j = 1}^{n} v^{\prime}_{ij} ;\quad i = 1,2, \ldots ,m$$

(7)

Here $\overline{{v^{\prime}_{j} }}$ presents the mean of the jth attribute and $\overline{{v^{\prime}_{k} }}$ presents the mean of the kth attribute, where $\overline{{v^{\prime}_{j} }}$ is computed from Eq. (7). $d_{ik}$ ranging from − 1 to 1 indicates the direction and strength of the relationship between criteria j and k.

Step 5 For each criterion compute conflict intensity.

By using the standard deviation and correlation data, calculate the conflict intensity for each criterion $j$:

$$C_{j} = \sigma_{j} \times \mathop \sum \limits_{k = 1}^{m} \left( {1 - d_{ik} } \right).$$

(8)

Conflict intensity will be higher for criteria with high variability and low redundancy (low correlation), which combines the criterion’s variability with its correlation to other criteria.

Step 6 Calculate criteria weights.

To determine the final weights $w_{j}$, normalize the conflict intensity value $C_{j}$:

$$w_{j} = \frac{{C_{j} }}{{\mathop \sum \nolimits_{j = 1}^{n} C_{j} }};\quad j = 1,2, \ldots ,n$$

(9)

In this scenario, the contribution or significance of the jth criterion is represented by $C_{j}$, which is derived from the correlation coefficients $d_{ik}$ between the criterion and others as well as its standard deviation $\sigma_{j}$. To determine their relative importance, the qualities are arranged in descending order according to their calculated weights. By giving factors with high variability (standard deviation) and little association with other criteria more weight, this method guarantees an impartial and fair evaluation of their impact. In our study, the importance of topological indices in the framework for decision-making was determined using the CRITIC approach. Table 17 shows each index’s computed weights, which represent the indices relative significance within the analysis. Figure 3 represents weighted values graphically.

Table 17 Weight calculated by CRITIC method.

Fig. 3

Visual line graph representation of weight values.

Methodology of CoCoSo and ranking analysis of alternatives

Using a set of standardised criteria created during our case study, we are evaluating several drug alternatives for treating PCOS (polycystic ovary syndrome). By using these criteria, we can collect accurate quantitative information on structure–property relationships (QSPR), which helps us determine the best course of action. Finding the best course of treatment while weighing safety, effectiveness, and other crucial considerations is our aim. When an ideal course of action is not possible, the CoCoSo (combined compromise solution) approach is used to rank the options and determine which one is closest to the best result. The CoCoSo framework first clearly defines the alternatives and criteria for evaluating drugs for the treatment of PCOS. The CoCoSo decision-making process’s following steps are then methodically implemented and verified, guaranteeing a solid and trustworthy analysis that aids in well-informed decision-making.

The following steps are followed when the alternatives and criteria have been established.

Step 1 Construct the decision matrix which is shown in Table 2.

Step 2 By the following equations the normalisation of the defined matrix $\left[ {v_{ij} } \right]_{m \times n}$.

$$nv^{\prime}_{ij} = \frac{{v_{ij} - minv_{ij} }}{{maxv_{ij} - minv_{ij} }}\quad {\text{for Beneficial Criterion}}$$

(10)

$$nv^{\prime}_{ij} = \frac{{maxv_{ij} - v_{ij} }}{{maxv_{ij} - minv_{ij} }}\quad {\text{for Non-Beneficial Criterion}}$$

(11)

where the normalised $nv^{\prime}_{ij}$ for alternative $i$ is determined with criterion $j$, $v_{ij}$ represents the rating of alternative $i$ in relation to criterion $j$. The results of normalised decision matrix are shown in Table 18.

Table 18 Normalised decision matrix.

Step 3 The total power weight of comparability sequences and the sum of weighted comparability sequences are represented by $Pw_{i}$ and $Sw_{i}$. They are calculated by following Eqs. (12) and (13), which are represented in Table 19. Tables 20 and 21 represent the results of the sum and power of the weighted comparability matrix.

$$Sw_{i} = \mathop \sum \limits_{j = 1}^{n} \left( {w_{j} \times nv_{ij} } \right)$$

(12)

$$Pw_{i} = \mathop \sum \limits_{j = 1}^{n} \left( {nv_{ij} } \right)^{{w_{j} }}$$

(13)

Table 19 calculated values of $Pw_{i}$ and $Sw_{i}$.

Table 20 Sum of the weighted comparability matrix.

Table 21 Power of the weighted comparability matrix.

Step 4 Using the relative performance score (R), the chosen alternatives are ranked as the last stage in the COCOSO approach. The combined estimated outcomes, represented as $R_{ix}$, $R_{iy}$ and $R_{iz}$, are carefully used to calculate this score. The COCOSO technique guarantees a more thorough and reliable ranking system by combining these findings. By smoothly combining the combined contributions of $R_{ix}$, $R_{iy}$ and $R_{iz}$, the computation of $R_{i}$ not only does the evaluation process but also the effectiveness of the approach increases. This methodical process enables a comprehensive evaluation, which eventually produces rankings that are trustworthy and perceptive. Table 22 shows the ranking of the drugs of PCOS by using Eqs. (14) to (17).

$$R_{i} = \left( {R_{ix} R_{iy} R_{iz} } \right) + \frac{1}{3}\left( {R_{ix} + R_{iy} + R_{iz} } \right)$$

(14)

$$R_{ix} = \frac{{S_{{w_{i} }} + Pw_{i} }}{{\mathop \sum \nolimits_{i = 1}^{m} \left( {S_{{w_{i} }} + Pw_{i} } \right)}},$$

(15)

$$R_{iy} = \frac{{S_{{w_{i} }} }}{{minS_{{w_{i} }} }} + \frac{{Pw_{i} }}{{minPw_{i} }},$$

(16)

$$R_{iz} + \frac{{\lambda S_{{w_{i} }} + \left( {1 - \lambda } \right)Pw_{i} }}{{\lambda maxS_{{w_{i} }} + \left( {1 - \lambda } \right)maxPw_{i} }},\quad 0 \le {\uplambda } \le 1$$

(17)

Table 22 By CoCoSo methodology ranking of the drugs.

Methodology of MABAC and ranking analysis of alternatives

The MABAC (multi-attributive border approximation area comparison) approach ranks and evaluates choices based on the distance each one is from an ideal border approximation area. It is most suited for rating PCOS drugs, considering several criteria like availability, cost, side effects, and efficacy. Through the integration of these elements, MABAC guarantees a fair and trustworthy ranking, assisting medical practitioners in choosing the best and most patient-centred treatment alternatives. The steps to determine the MABAC technique are given below:

Step 1 Same as CoCoSo method.

Step 2 Normalise the decision matrix.

For beneficial criterion:

$$nv^{\prime}_{ij} = \frac{{v_{ij} - minv_{i} }}{{maxv_{i} - minv_{i} }};\quad i = 1,2, \ldots ,m\; {\text{and}}\;j = 1,2, \ldots ,m$$

(18)

For non-beneficial criterion:

$$nv^{\prime}_{ij} = \frac{{maxv_{i} - v_{i} }}{{maxv_{i} - minv_{i} }};\quad i = 1,2, \ldots ,m \;{\text{and}}\;j = 1,2, \ldots ,m$$

(19)

where $nv^{\prime}_{ij}$ represents the normalise decision matrix of ith alternative in jth attribute. Table 18 shows normalised decision matrix.

Step 3 After assigning the weighted normalised values for each attribute are obtained using Eq. (20). Results of the weighted normalised matrix are shown in Table 23.

$$v^{\prime}_{ij = } W_{j} \left( {nv^{\prime}_{ij} + 1} \right).$$

(20)

Table 23 Weighted normalised matrix.

Step 4 The Border approximation area matrix is gained for Eq. (21) which are shown n Table 24.

$$h_{j} = \left( {\mathop \prod \limits_{i = 1}^{m} v^{\prime}_{ij} } \right)^{\frac{1}{m}} .$$

(21)

where m is number of alternatives.

Table 24 Border approximation area matrix.

Step 5 Equation (22) can be used to determine the distance of each alternative from the border approximation area by using the weighted normalised values and the values of the border approximation area matrix. Table 25 displays the computation results utilising this equation.

$$t_{ij} = v^{\prime}_{ij} - h_{j} .$$

(22)

Table 25 Distance from border approximation area matrix.

Step 6 Equation (23) is used to determine the total distances between each option and the approximate border area. Their final ranking, which is displayed in Table 26, is determined by using this calculation to rank them in descending order.

$$F_{i} = \mathop \sum \limits_{j = 1}^{n} t_{ij} .$$

(23)

Table 26 By MABAC method ranking of the drugs for PCOS.

Comparison of CoCoSo and MABAC drug ranking

CoCoSo and MABAC rankings for PCOS medications are being compared in order to assess how each approach ranks options according to a variety of factors, pointing out both the similarities and differences in the results.

After examining the findings, a noteworthy discovery was made. According to the CoCoSo and MABAC techniques, Orlistat was the best. According to the standards set for this assessment, this demonstrates the drug’s efficacy. Orlistat was in the top spot according to the CoCoSo technique, which combines weighted average and the total of relative performance as labelled in Table 27. Orlistat also maintained its outstanding performance by achieving the 1 rank in the MABAC technique. On the other hand, for both approaches, the drug finasteride was positioned second. For CoCoSo, spironolactone ranked third, and for MABAC, dexamethasone ranked third. In terms of results, dexamethasone fared the worst in CoCoSo, whereas Metformin had its worst showing in MABAC. A graph can be applied to visually compare the CoCoSo and MABAC methodology ranks and quickly display any parallels or discrepancies. A Graph representing the rankings of the drugs in CoCoSo and MABAC is shown in Fig. 4.

Table 27 Comparison of CoCoSo and MABAC drug ranking.

Fig. 4

Comparison of CoCoSo and MABAC methods for ranking PCOS drugs.

Discussion

In this study, a comprehensive evaluation of PCOS drugs was conducted using both QSPR analysis and multi-criteria decision-making (MCDM) techniques. The QSPR analysis was performed using degree-based and Banhatti indices to establish quantitative relationships between molecular descriptors and drug efficacy. The results from the linear regression model demonstrated a strong correlation, with correlation values ranging from 0.8 to 0.9, indicating a reliable predictive capability of the selected topological indices in assessing the physicochemical properties of the drugs. For drug ranking, two MCDM methods, CoCoSo and MABAC, were employed. Both methods consistently identified Orlistat as the most effective drug, highlighting its superior performance based on the applied criteria. The consistency in ranking across both techniques reinforces the robustness of the decision-making approach. Additionally, a comparative analysis of CoCoSo and MABAC rankings was presented in Table 27, ensuring a clear understanding of their methodological differences. Overall, the integration of QSPR analysis with MCDM techniques provided a comprehensive framework for evaluating PCOS drugs. The strong correlation in regression analysis validates the predictive reliability of topological indices, while the consistency in MCDM rankings further strengthens the credibility of the results. This combined approach enhances decision-making in drug selection, offering valuable insights for future pharmaceutical studies.

Conclusion

The presented article provides a comprehensive analysis of PCOS, including its aetiology and available treatments. We offer fresh perspectives on the molecular mechanisms behind PCOS and the pharmacological characteristics of potential treatments by fusing chemical graph theory with QSPR modelling. A data-driven approach to drug development has been made possible by the successful association between molecular structures and pharmacological efficacy established by the use of topological indices, such as the Banhatti and Zagreb indices. The evaluation and ranking of therapeutic alternatives have been further enhanced by the use of MCDM techniques; the CoCoSo and MABAC methods show that orlistat is the most effective medication. Of all the drugs, orlistat was ranked first. Using a linear regression model, the correlation (R) between all degree-based topological indices and physicochemical properties is greater than 0.9. For example, the modified first Banhatti index displays correlations (R) of (0.9322) for MV, (0.9082) for BP, (0.8519) for E, (0.9186) for MR, (0.8928) for P, (0.7032) for ACD/logP, and (0.9322) for MV. This study emphasises the important role of computational tools in tackling complex medical challenges. By connecting the fields of chemistry, medicine, and mathematics, it establishes a standard for further study in order to maximise PCOS treatment approaches.

In future research, the application of advanced topological indices, including fuzzy-based approaches, can be explored to handle uncertainty in molecular structures more effectively. Additionally, integrating other MCDM techniques or hybrid models may further enhance drug ranking accuracy. Extending this framework to a broader range of diseases and molecular datasets could provide deeper insights into the structure–activity relationship of pharmaceutical compounds.