A QSPR study of coronary artery disease drugs using eccentricity-based indices

Abstract

Quantitative structure-property relationship (QSPR) and Quantitative structure-activity relationship QSAR modeling are constructed on the principle, which describes that biological activity and physicochemical properties of a chemical compound can be deduced from its chemical structure. These relationships are commonly developed from graph invariants that can be computed from the molecular graphs of chemical compounds. Coronary artery disease occurs when the coronary arteries become narrowed or blocked, which can restrict the flow of oxygen-rich blood to the heart. Coronary artery disease is the leading cause of disability-adjusted life years lost and death worldwide. This study advances QSPR modeling by using eccentricity-based graphical invariants, specifically designed to enhance the predictive accuracy for physicochemical properties of drugs used to treat coronary artery disease, including atorvastatin, simvastatin, rosuvastatin, aspirin, clopidogrel, metoprolol, atenolol, enalapril, lisinopril, amlodipine, diltiazem, nitroglycerin, isosorbide dinitrate, ranolazine, gemfibrozil, and fenofibrate. We used the cubic, logarithmic, quadratic, and linear models to explore the structure-property relationship of drugs for coronary artery disease. We designed the models on the basis of the adjusted r-squared values, assuming the eccentricity-based invariants as independent variables, while the physico-chemical properties of sixteen drugs as dependent variables. The dependent variables include boiling point, enthalpy of vaporization, heavy atom count, molar volume, polarizability, complexity, molecular weight, and molar refractivity. The statistical analysis indicates that the most suitable structure-property models are nonlinear. The findings show that the eccentric Albertson index and the eccentric geometric arithmetic index attain superior predictive performance compared to other indices. The analysis shows that cubic regression is the optimal choice for predicting enthalpy of vaporization, molar refractivity, polarizability, and complexity. In contrast, quadratic regression is the best option for predicting molecular weight, while a linear model is most effective for assessing heavy atom count. Additionally, logarithmic regression is the most suitable choice for boiling point and molar volume. To validate the robustness of our regression models, we used them to assess the properties of five additional coronary artery disease drugs that were not part of the original dataset. The experimental values were compared with forecasted data, revealing a strong correlation between them. This demonstrates the reliability of our regression models in assessing these vital physicochemical parameters.

Introduction

Cheminformatics is a modern field involving Mathematics, Statistics, Physics, Biochemistry, Chemistry, Biology, and other informational sciences that concentrate on treating, analyzing, gathering, and storing chemical data. Cheminformatics is an influential application of information technology to help chemists analyze new problems as well as in understanding, investigating, and organizing scientific data to design unknown processes and compounds^1,2. Molecular and chemical graphs play a vital role in virtual screening, computational drug design, quantitative structure-activity (QSAR), quantitative structure-property (QSPR) relationship studies, and chemoinformatics. Chemoinformatics uses graphs for various applications, including database searching and retrieval, physicochemical property prediction, chemical structure illustration, and coding. QSAR study, virtual screening, and QSPR analysis are built on the structure-property principle, which expresses that physicochemical properties and biological activity of chemical compounds can be predicted based on their chemical structure. Such structure-property associations are commonly developed from graph topological indices, fingerprints computed from the molecular graph, and molecular descriptors summed from the chemical structure^3–6.

Chemical graph theory investigates the chemical and physical properties, including molar refraction, melting points, freezing points, boiling points, the octanol-water partition coefficient, the motor octane number, the acentric factor, surface tension and density, molar volume, heat of formation and solubility of the chemical graphs with the aid of mathematical models which are developed by the graph-theoretic techniques of graph theory. Cheminformatics approaches are also used to predict properties relevant to drug discovery and optimization. For example, knowledge discovery can be employed to determine lead mixtures in pharmaceutical data matching. This modern technological era concentrates on advancing chemical sciences, including designing diagnostic kits for different biological approaches and infections, drug discovery, and bond formation in chemical compounds. All these developments require powerful tools to make these advancements possible as they progress. An emerging tool for analyzing these phenomena is a topological index, which gives the same numerical value to two isomorphic chemical networks.

Numerous topological invariants are employed in QSPR/QSAR studies^3,5, supplying insights into the underlying structures. Different kinds of graph invariants used for this purpose include polynomial, distance, spectral, and degree-based invariants. Among the various classes, distance-based invariants are valuable as they are used to deduce the physicochemical, biological, and pharmacological properties including toxicity, similarity, connectivity, melting point, boiling point, entropy, chirality, surface tension, stability, density, critical temperature, molecular weight, molar refractivity, polarizability, molar volume and the enthalpy of vaporization^7–9. Kirmani et al.¹⁰ examined the degree-based topological invariants of COVID-19 medications, detecting good relationships with their properties. Liu et al.¹¹ computed the distance-based invariants for antiviral drugs. Havare¹² employed the M-polynomial approach to set up QSPR models for cancer drugs. Adnan et al.¹³ associated the degree-based invariants with properties of anti-tuberculosis drugs. Shanmukha et al.¹⁴ viewed anticancer drugs, establishing connections between topological invariants and chemical characteristics. Zaman et al.¹⁵ developed QSPR models to forecast molecular properties utilizing a linear regression model. Ravi and Desikan¹⁶ operated reduced reverse degree-based invariants to develop QSPR models for hyaluronic acid-paclitaxel conjugates, showing strong predictive power. Gnanaraj et al.¹⁷ executed QSPR analysis on NSAID drugs, while Nasir et al.¹⁸ discussed blood cancer compounds. Further, QSPR modelling has been applied to study the drugs of infertility treatment drugs¹⁹, antitumor²⁰, breast cancer²¹, anti-malarial²², Colorectal Cancer²³, cardiovascular disease²⁴, and HIV/AIDS²⁵. These studies shows that QSPR analysis furnishes a novel technique for predictive modeling, enriching our understanding of how topological invariants can improve drug development and drug design. However, much of the QSPR research has concentrated on degree-based molecular descriptors, creating a gap in the exploration of eccentricity-based descriptors. This study handles this gap by presenting these indices and showing their strong correlation with drug properties, presenting a novel technique to predictive modeling and improving the accuracy of drug design beyond existing approaches. Our research combines machine learning regression strategies with topological invariants from hydrogen-depleted molecular graphs, concentrating on eccentricity-based descriptors. Our purpose is to improve the molecular interactions and enhance predictive modeling, eventually refining the drug design precision. This may aid in developing more useful generalised drugs for the under-considered disease.

Coronary artery disease is described by the formation of atherosclerosis in the coronary arteries, which can sometimes be asymptomatic. Coronary heart disease, also commonly known as ischemic heart disease, includes conditions such as silent myocardial ischemia, acute coronary syndrome, and stable angina. Deaths related to coronary heart disease are largely due to coronary artery disease. Acute coronary syndrome is generally symptomatic and possesses conditions such as myocardial infarction and unstable angina. Coronary artery disease is the leading cause of disability-adjusted life years (DALYs) lost and death worldwide. This issue mainly affects low- and middle-income countries, resulting in about 129 million DALYs and 7 million deaths annually. In 2015, there were 8.9 million deaths and 164.0 million DALYs globally due to coronary artery disease. Survivors of myocardial infarction have a much higher risk of recurrence, and they have an annual mortality rate 5 to 6 times greater than that of people without coronary artery disease²⁶. Coronary artery disease is characterized by an insufficient supply of oxygen and blood to the myocardium. The symptoms occur due to the blockage of the coronary arteries, leading to a mismatch in the demand and supply of oxygen. Coronary artery disease is commonly caused by the formation of plaques in the lumen of coronary arteries that restrict blood flow. Coronary artery disease is the major cause of death worldwide. Coronary artery disease was an unusual cause of death at the start of the 20th century. Casualties due to coronary artery disease increased in the mid-1960s and then decreased. However, this sickness is still the leading cause of death worldwide²⁷. The chance of death from coronary artery disease grows significantly with age. The modifiable risk factors include systolic blood pressure, which plays a vital role, as it contributes to the increased threat of coronary artery disease linked with ageing^28–30. The following Table 1 shows some drugs that are commonly used to treat coronary artery disease and their molecular graphs are displayed in Fig. 1.

Table 1.

Coronary artery disease drugs with corresponding chemical structures and therapeutic uses.

Drug name	Chemical structure	Therapeutic uses for coronary artery disease
Atorvastatin		Lowers cholesterol and prevents cardiovascular events by inhibiting HMG-CoA reductase, reducing LDL and triglycerides31.
Simvastatin		Lowers cholesterol and cardiovascular risk by blocking HMG-CoA reductase, reducing LDL and triglycerides32.
Rosuvastatin		Lowers cholesterol and cardiovascular risk by inhibiting HMG-CoA reductase33.
Aspirin		Prevents heart attacks and strokes via antiplatelet effects. Long-term use risks GI irritation and bleeding34,35.
Clopidogrel		Antiplatelet drug prevents cardiovascular events in atherosclerosis and post-stenting by inhibiting receptor36.
Metoprolol		-blocker treats angina, hypertension, arrhythmias; lowers heart rate, contractility, and oxygen demand37.
Atenolol		It is selective -blocker for hypertension, angina, and post- prevention; reduces heart rate and oxygen demand38.
Enalapril		ACE inhibitor treats hypertension and heart failure; inhibits angiotensin II conversion, causing vasodilation39.
Lisinopril		Long-acting ACE inhibitor for hypertension, heart failure, and post-myocardial infarction; reduces blood pressure and cardiac workload40.
Amlodipine		it treats for angina and hypertension; causes vasodilation41.
Diltiazem		it treats angina, hypertension, and arrhythmias; reduces contractility and causes vasodilation42.
Nitroglycerin		Nitrate vasodilator treats angina and heart failure; releases NO to relax vessels and improve coronary blood flow43,44.
Isosorbide dinitrate		Nitrate for angina prevention; vasodilates via NO release, reducing cardiac workload45.
Ranolazine		Treats chronic angina by blocking late sodium current, reducing calcium overload and oxygen demand46.
Gemfibrozil		Fibrate for hyperlipidemia; lowers triglycerides and raises HDL to reduce cardiovascular risk47.
Fenofibrate		Fibrate treats hypertriglyceridemia and hyperlipidemia, lowering cardiovascular risk, especially in metabolic syndrome48.

Fig. 1.

Coronary artery disease drugs: (a) Atorvastatin, (b) Simvastatin, (c) Rosuvastatin, (d) Aspirin, (e) Clopidogrel, (f) Metoprolol, (g) Atenolol, (h) Enalapril, (i) Lisinopril, (j) Amlodipine, (k) Diltiazem, (l) Nitroglycerin, (m) Isosorbide dinitrate, (n) Ranolazine, (o) Gemfibrozil, (p) Fenofibrate.

Materials and methods

This study begins by gathering data on the physicochemical properties of drugs used for coronary artery disease. The physico-chemical properties for drug compounds are displayed in the Table 2, which are gathered from ChemSpider⁴⁹ and PubChem⁵⁰.

Table 2.

Physico-chemical properties of drugs.

Drugs	MW (g/mol)	HAC	CO	BP	ENV (kj/mol)	MR	PO	MV
Atorvastatin	558.6	41	822	722.2	110.7	155.2	61.5	451.9
Simvastatin	418.6	30	706	564.9	97.5	116.4	46.1	376.6
Rosuvastatin	481.5	33	767	745.6	114.1	120.0	47.6	351.9
Aspirin	180.16	13	212	321.4	59.5	44.5	17.7	139.6
Clopidogrel	321.8	21	381	423.7	67.8	85.5	33.9	244.3
Metoprolol	267.36	19	215	398.6	68.5	77.1	30.6	258.7
Atenolol	266.34	19	263	508.0	82.0	74.3	29.4	236.7
Enalapril	376.4	27	519	582.4	91.6	99.5	39.5	312.6
Lisinopril	405.5	29	550	666.4	102.9	107.5	42.6	323.9
Amlodipine	408.9	28	647	527.2	80.2	105.4	41.8	333.0
Diltiazem	414.5	29	565	594.4	88.6	115.2	45.7	327.6
Nitroglycerin	227.09	15	218	295.8	51.4	39.2	15.5	135.9
Isosorbide dinitrate	236.14	16	268	365.9	58.8	44.7	17.7	142.9
Ranolazine	427.5	31	531	624.1	97.2	122.1	48.4	364.0
Gemfibrozil	250.33	18	273	394.7	68.0	71.9	28.5	239.7
Fenofibrate	360.8	25	458	469.8	73.2	97.1	38.5	306.4

The methodology of this study assumes an organized strategy to analyze and forecast the physicochemical properties of coronary artery disease drugs through QSPR modeling, using eccentricity-based topological indices. We develop linear, quadratic, logarithmic, and cubic regression models to correlate the physicochemical properties of sixteen coronary artery disease drugs structures by considering eccentricity-based topological indices as independent variables and physicochemical properties as dependent variables. The linear, logarithmic, quadratic, and cubic regression models have the following mathematical expressions.

1

2

3

4

In the above Equations ((1)-(4)), denotes the physico-chemical property, and is the topological index value of the underlying structure. The constants , and are the regression coefficients and is a constant.

Eccentricity-based topological invariants

Eccentricity-based topological invariants are essential for studying molecular structures, especially in network analysis and mathematical chemistry⁵¹ as they offer important information about molecular branching, stability, and general connectivity by encapsulating the extremal distances within a molecular framework^52,53. The eccentricity of a vertex r in a molecular graph is represented as the greatest shortest path distance of r to any other vertex in the graph, which is the source of these indices. We denote the eccentricity of a vertex r with We will now discuss several indices based on eccentricity that we utilize in our QSPR modeling. The eccentric version of first Zagreb index of a graph was proposed by Vukičević and Graovac in⁵⁴ as:

Ghorbani and Hosseinzadeh⁵⁵ introduced the second eccentric Zagreb index, which can be written as follows:

In⁵⁶, Ghorbani and Khaki defined the eccentric geometric arithmetic index of as:

The eccentric atom bond connectivity index was first given in⁵⁷ as:

The concept of sum-connectivity index was given in²⁵, which is defined as:

The Zaman et al. in²⁵ gave the idea of the eccentricity inverse sum and eccentricity Albertson index, their mathematical expressions are as follows:

Calculation of topological invariants

The eccentricity-based edge partitioning for the drugs of coronary artery disease is displayed in Tables 3 and 4. By using this edge partitioning, we aim to compute the eccentricity-based topological indices in this section. Let , . . . , be the molecular graphs of atorvastatin, simvastatin, rosuvastatin, aspirin, clopidogrel, metoprolol, atenolol, enalapril, lisinopril, amlodipine, diltiazem, nitroglycerin, isosorbide dinitrate, ranolazine, gemfibrozil, and fenofibrate respectively.

Table 3.

Edge partitioning of drug compounds based on eccentricities.

Edges	Atorvastatin	Simvastatin	Rosuvastatin	Aspirin	Clopidogrel	Metoprolol	Atenolol	Enalapril	Lisinopril
(16,15)	4	0	0	0	0	0	0	0	0
(15,14)	3	0	0	0	0	0	0	0	0
(14,13)	7	0	6	0	0	3	0	0	0
(13,12)	6	5	5	0	0	2	4	4	5
(12,12)	0	0	0	0	0	0	0	1	1
(12,11)	8	5	5	0	0	2	2	5	6
(11,10)	7	5	7	0	0	3	2	5	5
(10,10)	1	0	0	0	0	0	0	0	0
(10,9)	5	4	6	0	0	3	4	4	4
(9,9)	0	1	0	0	1	0	0	0	0
(9,8)	3	7	3	0	4	3	3	5	5
(8,7)	0	4	2	0	7	3	3	3	3
(7,7)	0	1	0	0	0	0	1	1	1
(7,6)	0	0	0	0	6	0	0	0	0
(6,5)	0	0	0	6	4	0	0	0	0
(5,5)	0	0	0	0	1	0	0	0	0
(5,4)	0	0	0	4	0	0	0	0	0
(4,3)	0	0	0	3	0	0	0	0	0

Table 4.

Edge partitioning of drug compounds based on eccentricities.

Edges	Amlodipine	Diltiazem	Nitroglycerin	Isosorbide dinitrate	Ranolazine	Gemfibrozil	Fenofibrate
(18,17)	0	0	0	0	5	0	0
(17,16)	0	0	0	0	7	0	0
(16,15)	0	0	0	0	4	0	0
(15,14)	0	0	0	0	2	0	3
(14,13)	0	0	0	0	3	0	3
(13,12)	0	3	0	0	3	0	4
(12,11)	0	2	0	0	2	0	5
(11,10)	4	5	0	0	3	0	3
(10,9)	7	7	0	0	4	5	2
(9,9)	0	1	0	0	0	0	0
(9,8)	5	7	0	4	0	6	4
(8,8)	0	0	0	0	0	0	2
(8,7)	7	4	4	2	0	3	0
(7,7)	1	2	0	0	0	0	0
(7,6)	4	0	4	2	0	2	0
(6,6)	1	0	0	2	0	0	0
(6,5)	0	0	3	4	0	2	0
(5,5)	0	0	0	3	0	0	0
(5,4)	0	0	3	0	0	0	0

To show the computation of eccentricity-based indices, we considered ranolazine () as the reference structure.

• i.
  By using the first and second eccentric Zagreb indices and Table 4, we have

  
• ii.
  By using the eccentric geometric arithmetic index and Table 4, we have

  

• iii
  By using the eccentric atom bond connectivity index and Table 4, we have

  
• iv
  By using the eccentric sum-connectivity index and Table 4, we have

  
• v
  By using the eccentric ISI index and Table 4, we have

  
• vi
  By using the eccentricity Albertson index and Table 4, we have

  

The values of eccentricity-based topological invariants for other medications are shown in the Tables 5 and 6.

Table 5.

Eccentricity based topological indices for drug molecules.

Drug molecules
Atorvastatin	1047	6392	43.958899	17.431132	9.116591	261.286868	43
Simvastatin	632	3208	31.959034	13.792269	7.284833	157.613889	30
Rosuvastatin	744	4170	33.961252	13.987772	7.345297	185.600365	34
Aspirin	123	296	12.919617	7.589259	4.276295	30.395382	13
Clopidogrel	323	1158	22.943160	11.470044	6.199611	80.368216	21
Metoprolol	393	2106	18.975020	8.042943	4.239484	98.011177	19
Atenolol	374	1901	18.975226	8.212897	4.340592	93.267702	18
Enalapril	564	2915	27.965522	11.960036	6.309985	140.669967	26

Table 6.

Eccentricity based topological indices for drug molecules.

Drug molecules
Lisinopril	612	3203	29.963776	12.742872	6.718499	152.649098	28
Amlodipine	485	2075	28.949677	13.428558	7.164189	120.843157	27
Diltiazem	584	2819	30.959314	13.626448	7.214798	145.627024	28
Nitroglycerin	172	542	13.948251	7.392305	4.046730	42.704895	14
Isosorbide dinitrate	222	751	16.966135	8.754938	4.773319	55.278473	12
Ranolazine	929	6782	32.976175	12.159509	6.320205	231.943893	33
Gemfibrozil	290	1194	17.961799	8.476094	4.534612	72.212059	18
Fenofibrate	584	3386	25.975126	10.594982	5.560636	145.731159	24

Results and discussions

To determine the most suitable topological index for forecasting molecular properties, we perform regression analysis using the R language and SPSS, employing cubic, linear, logarithmic, and quadratic models. For each index, we evaluate the statistical significance of the models by examining metrics such as p-values, standard error, F-statistic, , and adjusted . We aim to specify the model that provides the most robust forecasting. The best-performing model is validated by comparing predicted versus actual values. Our selection criteria focus on achieving the highest adjusted values, the lowest standard error, and statistical effectiveness with . This approach offers a data-driven strategy that strikes a balance between clarity and accuracy.

Analysis of regression models for the molecular weight

Table 7 displays the values and adjusted values of cubic, quadratic, logarithmic, and linear regression models by assuming the sets of values of eccentricity-based invariants as independent variables and the set of values of molecular weight of sixteen drugs as the dependent variable. Figure 2 illustrates a graphical comparison of the adjusted values for these regression models. When we analyse the adjusted values given in Table 7, it is easy to observe that and invariants provide the best values within the linear model, while the other invariants achieve their most suitable fits in the quadratic model. Table 8 displays the statistical analysis of the best regression models for each independent variable. This analysis possesses the p value, standard error, F-statistic, correlation coefficient r, , adjusted , and regression equation. This statistical examination signifies that the eccentric geometric arithmetic index is the most predictive index utilizing a quadratic model.

Table 7.

Comparison of and adjusted -values for different regression models for MW.

Topological	Linear		Logarithmic		Quadratic		Cubic
Index		Adjusted		Adjusted		Adjusted		Adjusted
	0.874299	0.865320	0.856989	0.846774	0.890339	0.873469	0.890459	0.863073
	0.724687	0.705021	0.772221	0.755951	0.810052	0.780830	0.822879	0.778599
	0.985043	0.983974	0.967256	0.964917	0.986685		0.987197	0.983997
	0.933879	0.929156	0.927867	0.922715	0.934719	0.924675	0.935251	0.919064
	0.907758	0.901169	0.900360	0.893243	0.908118	0.893982	0.908301	0.885377
	0.873845	0.864834	0.856012	0.845728	0.889872	0.872929	0.889983	0.862479
	0.962844	0.960189	0.942226	0.938099	0.965824	0.960566	0.968124	0.960155

Fig. 2.

Comparison of adjusted -values.

Table 8.

Best regression models for MW based on adjusted values.

Predictor		Goodness of Fit				Significance		Regression Equation
Index	Model	r		Adj.	Std. Error	F-statistic	p-value
	Quadratic	0.993320	0.986685	0.984636	12.980935	4.82e+02	6.43e-13
	Quadratic	0.982764	0.965824	0.960566	20.796562	1.84e+02	2.95e-10
	Linear	0.966374	0.933879	0.929156	27.874683	1.98e+02	1.19e-09
	Linear	0.952763	0.907758	0.901169	32.923394	1.38e+02	1.24e-08
	Quadratic	0.943578	0.890339	0.873469	37.252667	5.28e+01	5.76e-07
	Quadratic	0.943330	0.889872	0.872929	37.332017	5.25e+01	5.92e-07
	Quadratic	0.900029	0.810052	0.780830	49.028568	2.77e+01	2.05e-05	(8e-06)

The comparison of the actual and the predicted values of molecular weights of drugs by using the best forecasted index is illustrated in Table 9. The related graphical comparison is displayed in Fig. 3.

Table 9.

Predicted vs Actual values for MW using the overall best model.

Sample	Actual Value	Predicted Value	Residual	Sample	Actual Value	Predicted Value	Residual
	558.60	562.3052	−3.705164		405.50	406.5006	−1.000645
	418.60	429.9467	−11.346731		408.90	394.4269	14.473137
	481.50	453.0623	28.437686		414.50	418.2504	−3.750384
	180.16	189.5003	−9.340299		227.09	203.4451	23.644881
	321.80	320.7415	1.058469		236.14	243.7284	−7.588429
	267.36	270.0232	−2.663248		427.50	441.7412	−14.241224
	266.34	270.0259	−3.685923		250.33	256.8129	−6.482901
	376.40	382.6083	−6.208289		360.80	358.4009	2.399063

Fig. 3.

Actual and predicted values for molecular weight.

Analysis of regression models for the heavy atom count

Table 10 provides the values and adjusted values of cubic, quadratic, logarithmic, and linear regression models by assuming the sets of values of eccentricity-based invariants as independent variables and the set of values of heavy atom count of sixteen drugs as the dependent variable. Figure 4 illustrates a graphical comparison of the adjusted values for these regression models. When we analyse the adjusted values given in Table 10, it is easy to observe that , and invariants provide the best values within the quadratic model, while the other invariants achieve their most suitable fits in the linear model. Table 11 displays the statistical analysis of the best regression models for each independent variable. This analysis possesses the p value, standard error, F-statistic, correlation coefficient r, , adjusted , and regression equation. This statistical examination shows that the eccentric geometric arithmetic index is the most predictive index utilizing a linear model.

Table 10.

Comparison of and adjusted -values for different regression models for HAC.

Topological	Linear		Logarithmic		Quadratic		Cubic
Index		Adjusted		Adjusted		Adjusted		Adjusted
	0.912235	0.905966	0.871588	0.862415	0.920070	0.907773	0.920436	0.900545
	0.772895	0.756673	0.793271	0.778505	0.842372	0.818122	0.852872	0.816090
	0.994521		0.962680	0.960014	0.994528	0.993686	0.994817	0.993521
	0.919402	0.913645	0.904109	0.897259	0.919649	0.907287	0.921551	0.901939
	0.890092	0.882241	0.874160	0.865172	0.890751	0.873944	0.892221	0.865276
	0.911825	0.905526	0.870540	0.861293	0.919653	0.907292	0.920004	0.900005
	0.980917	0.979554	0.948883	0.945232	0.981329	0.978456	0.982312	0.977890

Fig. 4.

Comparison of adjusted -values for heavy atom count.

Table 11.

Best regression models for HAC based on adjusted values.

Predictor		Goodness of Fit				Significance		Regression Equation
Index	Model	r		Adj.	Std. Error	F-statistic	p-value
	Linear	0.997257	0.994521	0.994130	0.588763	2541.230259	2.20e-16
	Linear	0.990413	0.980917	0.979554	1.098784	719.646188	1.95e-13
	Linear	0.958855	0.919402	0.913645	2.258151	159.702170	4.80e-09
	Quadratic	0.959203	0.920070	0.907773	2.333662	74.821510	7.37e-08	(9e-06)
	Quadratic	0.958986	0.919653	0.907292	2.339741	74.399489	7.63e-08
	Linear	0.943447	0.890092	0.882241	2.636980	113.378882	4.27e-08
	Quadratic	0.917808	0.842372	0.818122	3.277180	34.736386	6.09e-06	(1e-06)

The comparison of the actual and the predicted values of heavy atom count of drugs by using the best forecasted index is illustrated in Table 12. The related graphical comparison is displayed in Fig. 5.

Table 12.

Predicted vs Actual values for HAC using the overall best model.

Sample	Actual Value	Predicted Value	Residual	Sample	Actual Value	Predicted Value	Residual
	41	41.13679	−0.136790		29	28.56009	0.439906
	30	30.35313	−0.353129		28	27.64878	0.351225
	33	32.15242	0.847580		29	29.45473	−0.454733
	13	13.24339	−0.243386		15	14.16777	0.832234
	21	22.25103	−1.251028		16	16.87978	−0.879783
	19	18.68507	0.314935		31	31.26718	−0.267182
	19	18.68525	0.314750		18	17.77454	0.225465
	27	26.76437	0.235634		25	24.97570	0.024301

Fig. 5.

Actual and predicted values for heavy atom count.

Analysis of regression models for the complexity

Table 13 provides the values and adjusted values of cubic, quadratic, logarithmic, and linear regression models by assuming the sets of values of eccentricity-based invariants as independent variables and the set of values of complexity of sixteen drugs as the dependent variable. Figure 6 illustrates a graphical comparison of the adjusted values for these regression models. When we analyse the adjusted values given in Table 13, it is easy to observe that and invariants provide the best values within the quadratic model, while the other invariants achieve their most suitable fits in the cubic model. Table 14 displays the statistical analysis of the best regression models for each independent variable. This analysis possesses the p value, standard error, F-statistic, correlation coefficient r, , adjusted , and regression equation. This statistical examination tells us that the eccentric atom bond connectivity index is the most predictive index utilizing a cubic model.

Table 13.

Comparison of and adjusted -values for different regression models for CO.

Topological	Linear		Logarithmic		Quadratic		Cubic
Index		Adjusted		Adjusted		Adjusted		Adjusted
	0.714386	0.693985	0.693771	0.671898	0.736900	0.696423	0.749029	0.686286
	0.546616	0.514231	0.590441	0.561187	0.654342	0.601164	0.712258	0.640322
	0.902905	0.895969	0.878777	0.870118	0.905060	0.890454	0.928945	0.911182
	0.937246	0.932764	0.933147	0.928372	0.940206	0.931007	0.952524
	0.928776	0.923689	0.927047	0.921836	0.932090	0.921643	0.940695	0.925869
	0.713747	0.693301	0.692581	0.670622	0.736236	0.695657	0.748462	0.685578
	0.867144	0.857654	0.840637	0.829253	0.870616	0.850711	0.899973	0.874966

Fig. 6.

Comparison of adjusted -values for complexity.

Table 14.

Best regression models for CO based on adjusted values.

Predictor		Goodness of Fit				Significance		Regression Equation
Index	Model	r		Adj.	Std. Error	F-statistic	p-value
	Cubic	−0.975973	0.952524	0.940655	50.528602	80.252925	3.29e-08
	Cubic	−0.969894	0.940695	0.925869	56.473556	63.448032	1.24e-07
	Cubic	−0.963818	0.928945	0.911182	61.815260	52.294782	3.66e-07
	Cubic	−0.948669	0.899973	0.874966	73.342963	35.989194	2.81e-06
	Quadratic	0.858429	0.736900	0.696423	114.282391	18.205416	1.70e-04
	Quadratic	0.858042	0.736236	0.695657	114.426544	18.143208	1.73e-04
	Cubic	−0.843954	0.712258	0.640322	124.394602	9.901334	0.001444

The comparison of the actual and the predicted values of complexity of drugs by using the best forecasted index is illustrated in Table 15. The related graphical comparison is displayed in Fig. 7.

Table 15.

Predicted vs Actual values for CO using the overall best model.

Sample	Actual Value	Predicted Value	Residual	Sample	Actual Value	Predicted Value	Residual
	822	828.9449	−6.944899		550	579.4878	−29.487787
	706	668.2950	37.705025		647	638.5343	8.465707
	767	683.6907	83.309348		565	654.8937	−89.893685
	212	224.8317	−12.831685		218	221.6271	−3.627067
	381	466.2723	−85.272284		268	266.5948	1.405189
	215	236.6888	−21.688792		531	527.6011	3.398949
	263	242.6386	20.361353		273	253.3600	19.639958
	519	509.7713	9.228716		458	391.7680	66.231954

Fig. 7.

Actual and predicted values for complexity.

Analysis of regression models for the boiling point

Table 16 provides the values and adjusted values of cubic, quadratic, logarithmic, and linear regression models by assuming the sets of values of eccentricity-based invariants as independent variables and the set of values of boiling point of sixteen drugs as the dependent variable. Figure 8 illustrates a graphical comparison of the adjusted values for these regression models. When we analyse the adjusted values given in Table 16, we see that and provide best values within the quadratic model, and give best values within the cubic model, while the other invariants achieve their most suitable fits in the logarithmic model. Table 17 displays the statistical analysis of the best regression models for each independent variable. This analysis possesses the p value, standard error, F-statistic, correlation coefficient r, , adjusted , and regression equation. This statistical examination signifies that the eccentric geometric arithmetic index is the most predictive index utilizing a logarithmic model.

Table 16.

Comparison of and adjusted -values for different regression models for BP.

Topological	Linear		Logarithmic		Quadratic		Cubic
Index		Adjusted		Adjusted		Adjusted		Adjusted
	0.803841	0.789830	0.815672	0.802506	0.845792	0.822068	0.852620	0.815775
	0.680966	0.658178	0.760167	0.743037	0.813062	0.784302	0.830574	0.788217
	0.848465	0.837642	0.853974		0.861418	0.840098	0.867917	0.834896
	0.750877	0.733082	0.760776	0.743688	0.760995	0.724225	0.761306	0.701633
	0.718354	0.698236	0.725830	0.706247	0.726539	0.684468	0.727312	0.659140
	0.803609	0.789581	0.814910	0.801689	0.845548	0.821786	0.852449	0.815561
	0.842582	0.831338	0.842332	0.831070	0.856414	0.834324	0.872757	0.840946

Fig. 8.

Comparison of adjusted -values for boiling point.

Table 17.

Best regression models for BP based on adjusted values.

Predictor		Goodness of Fit				Significance		Regression Equation
Index	Model	r		Adj.	Std. Error	F-statistic	p-value
	Logarithmic	0.924107	0.853974	0.843543	55.050756	81.873217	3.18e-07
	Cubic	−0.934215	0.872757	0.840946	55.505753	27.435952	1.17e-05
	Quadratic	0.919670	0.845792	0.822068	58.707505	35.650878	5.28e-06
	Quadratic	0.919537	0.845548	0.821786	58.754019	35.584165	5.34e-06
	Cubic	0.911358	0.830574	0.788217	64.048857	19.609123	6.41e-05
	Logarithmic	0.872225	0.760776	0.743688	70.461293	44.522472	1.06e-05
	Logarithmic	0.851957	0.725830	0.706247	75.432358	37.063256	2.80e-05

The comparison of the actual and the predicted values of boiling point of drugs by using the best forecasted index is illustrated in Table 18. The related graphical comparison is displayed in Fig. 9.

Table 18.

Predicted vs Actual values for BP using the overall best model.

Sample	Actual Value	Predicted Value	Residual	Sample	Actual Value	Predicted Value	Residual
	722.2	731.7384	−9.538355		666.4	591.0799	75.320112
	564.9	614.7388	−49.838784		527.2	578.4440	−51.244037
	745.6	637.0396	108.560358		594.4	603.0752	−8.675165
	321.4	282.3442	39.055790		295.8	310.4591	−14.659054
	423.7	493.1022	−69.402224		365.9	382.3415	−16.441513
	398.6	423.4103	−24.810276		624.1	626.2370	−2.137044
	508.0	423.4143	84.585740		394.7	403.2707	−8.570745
	582.4	565.7507	16.649253		469.8	538.6541	−68.854056

Fig. 9.

Actual and predicted values for boiling point.

Analysis of regression models for the enthalpy of vaporization

Table 19 gives the values and adjusted values of cubic, quadratic, logarithmic, and linear regression models by assuming the sets of values of eccentricity-based invariants as independent variables and the set of values of enthalpy of vaporization of sixteen drugs as the dependent variable. Figure 10 illustrates a graphical comparison of the adjusted values for these regression models. When we analyse the adjusted values given in Table 19, we see that provides best value within the linear model, and provide best values within the logarithmic model, while the other invariants achieve their most suitable fits in the cubic model. Table 20 displays the statistical analysis of the best regression models for each independent variable. This analysis possesses the p value, standard error, F-statistic, correlation coefficient r, , adjusted , and regression equation. This statistical examination tells us that the eccentric Albertson index is the most predictive index utilizing a cubic model.

Table 19.

Comparison of and adjusted -values for different regression models for ENV.

Topological	Linear		Logarithmic		Quadratic		Cubic
Index		Adjusted		Adjusted		Adjusted		Adjusted
	0.788437	0.773325	0.783623	0.768168	0.822969	0.795734	0.836827	0.796034
	0.673627	0.650315	0.733567	0.714536	0.800774	0.770124	0.827186	0.783983
	0.817257	0.804204	0.811346	0.797871	0.824070	0.797004	0.837810	0.797263
	0.712015	0.691445	0.714198	0.693784	0.716535	0.672925	0.720105	0.650131
	0.679569	0.656681	0.680322	0.657488	0.682991	0.634220	0.687486	0.609358
	0.788191	0.773061	0.782728	0.767209	0.822704	0.795428	0.836688	0.795859
	0.833111	0.821190	0.829741	0.817579	0.844390	0.820450	0.862343

Fig. 10.

Comparison of adjusted -values for enthalpy of vaporization.

Table 20.

Best regression models for ENV based on adjusted values.

Predictor		Goodness of Fit				Significance		Regression Equation
Index	Model	r		Adj.	Std. Error	F-statistic	p-value
	Cubic	−0.928624	0.862343	0.827929	7.971441	25.057825	1.87e-05
	Linear	0.904023	0.817257	0.804204	8.503258	62.610334	1.56e-06
	Cubic	0.914783	0.836827	0.796034	8.678847	20.513914	5.13e-05
	Cubic	0.914706	0.836688	0.795859	8.682565	20.492929	5.16e-05
	Cubic	0.909498	0.827186	0.783983	8.931564	19.146311	7.21e-05
	Logarithmic	0.845103	0.714198	0.693784	10.634015	34.985045	3.77e-05
	Logarithmic	0.824816	0.680322	0.657488	11.246601	29.794084	8.43e-05

The comparison of the actual and the predicted values of enthalpy of vaporization of drugs by using the best forecasted index is illustrated in Table 21. The related graphical comparison is displayed in Fig. 11.

Table 21.

Predicted vs Actual values for ENV using the overall best model.

Sample	Actual Value	Predicted Value	Residual	Sample	Actual Value	Predicted Value	Residual
	110.7	111.54190	−0.841905		102.9	92.08564	10.814355
	97.5	97.15986	0.340135		80.2	89.44396	−9.243956
	114.1	105.83614	8.263859		88.6	92.08564	−3.485645
	59.5	58.64489	0.855105		51.4	59.77348	−8.373478
	67.8	73.55967	−5.759672		58.8	57.79462	1.005377
	68.5	68.79664	−0.296644		97.2	103.90784	−6.707836
	82.0	66.61697	15.383031		68.0	66.61697	1.383031
	91.6	86.76281	4.837189		73.2	81.37295	−8.172946

Fig. 11.

Actual and predicted values for enthalpy of vaporization.

Analysis of regression models for the molar refractivity

Table 22 provides the values and adjusted values of cubic, quadratic, logarithmic, and linear regression models by assuming the sets of values of eccentricity-based invariants as independent variables and the set of values of molar refractivity of sixteen drugs as the dependent variable. Figure 12 illustrates a graphical comparison of the adjusted values for these regression models. When we analyse the adjusted values given in Table 22, we see that provides best value within the quadratic model, gives best value within the linear model, has the best value within the logarithmic model, while the other invariants achieve their most suitable fits in the cubic model. Table 23 displays the statistical analysis of the best regression models for each independent variable. This analysis possesses the p value, standard error, F-statistic, correlation coefficient r, , adjusted , and regression equation. This statistical examination signifies that the eccentric Albertson index is the most predictive index utilizing a cubic model.

Table 22.

Comparison of and adjusted -values for different regression models for MR.

Topological	Linear		Logarithmic		Quadratic		Cubic
Index		Adjusted		Adjusted		Adjusted		Adjusted
	0.897725	0.890419	0.906348	0.899659	0.924093	0.912415	0.931142	0.913928
	0.765487	0.748736	0.839585	0.828127	0.860802	0.839387	0.860990	0.826237
	0.960663	0.957854	0.961736	0.959003	0.967005	0.961928	0.969759	0.962199
	0.868511	0.859119	0.869037	0.859683	0.870558	0.850643	0.879394	0.849243
	0.835144	0.823369	0.831532	0.819499	0.835845	0.810591	0.840504	0.800630
	0.897364	0.890033	0.905364	0.898604	0.923683	0.911942	0.930630	0.913287
	0.961233	0.958463	0.977055	0.975416	0.977164	0.973650	0.982530

Fig. 12.

Comparison of adjusted -values for Molar refractivity.

Table 23.

Best regression models for MR based on adjusted values.

Predictor		Goodness of Fit				Significance		Regression Equation
Index	Model	r		Adj.	Std. Error	F-statistic	p-value
	Cubic	0.991227	0.982530	0.978163	4.762799	224.970112	8.27e-11
	Cubic	0.984763	0.969759	0.962199	6.266424	128.270733	2.21e-09
	Cubic	0.964957	0.931142	0.913928	9.455796	54.090772	3.03e-07
	Cubic	0.964691	0.930630	0.913287	9.490926	53.661534	3.17e-07
	Logarithmic	0.932222	0.869037	0.859683	12.073184	92.900801	1.47e-07
	Quadratic	0.927794	0.860802	0.839387	12.916869	40.196041	2.71e-06	(2e-06)
	Linear	0.913862	0.835144	0.823369	13.545661	70.922783	7.49e-07

The comparison of the actual and the predicted values of molar refractivity of drugs by using the best forecasted index is illustrated in Table 24. The related graphical comparison is displayed in Fig. 13.

Table 24.

Predicted vs Actual values for MR using the overall best model.

Sample	Actual Value	Predicted Value	Residual	Sample	Actual Value	Predicted Value	Residual
	155.2	154.22907	0.970928		107.5	108.57082	−1.070823
	116.4	114.20733	2.192674		105.4	105.63813	−0.238130
	120.0	125.14759	−5.147589		115.2	108.57082	6.629177
	44.5	44.18539	0.314611		39.2	50.46877	−11.268771
	85.5	85.12853	0.371465		44.7	37.50657	7.193431
	77.1	76.68259	0.417409		122.1	122.40208	−0.302081
	74.3	72.06419	2.235807		71.9	72.06419	−0.164193
	99.5	102.60139	−3.101392		97.1	96.13252	0.967477

Fig. 13.

Actual and predicted values for Molar refractivity.

Analysis of regression models for the polarizability

Table 25 provides the values and adjusted values of cubic, quadratic, logarithmic, and linear regression models by assuming the sets of values of eccentricity-based invariants as independent variables and the set of values of polarizability of sixteen drugs as the dependent variable. Figure 14 illustrates a graphical comparison of the adjusted values for these regression models. When we analyse the adjusted values given in Table 25, we see that provides best value within the quadratic model, gives best value within the linear model, has the best value within the logarithmic model, while the other invariants achieve their most suitable fits in the cubic model. Table 26 displays the statistical analysis of the best regression models for each independent variable. This analysis possesses the p value, standard error, F-statistic, correlation coefficient r, , adjusted , and regression equation. This statistical examination signifies that the eccentric Albertson index is the most predictive index utilizing a cubic model.

Table 25.

Comparison of and adjusted -values for different regression models for PO.

Topological	Linear		Logarithmic		Quadratic		Cubic
Index		Adjusted		Adjusted		Adjusted		Adjusted
	0.897353	0.890021	0.905979	0.899263	0.923848	0.912133	0.930764	0.913455
	0.765103	0.748325	0.839134	0.827644	0.860636	0.839195	0.860811	0.826013
	0.960430	0.957603	0.961601	0.958859	0.966848	0.961748	0.969535	0.961919
	0.868466	0.859071	0.869129	0.859781	0.870573	0.850661	0.879358	0.849198
	0.835155	0.823380	0.831691	0.819669	0.835895	0.810648	0.840544	0.800680
	0.896991	0.889633	0.904991	0.898205	0.923437	0.911658	0.930250	0.912813
	0.961100	0.958322	0.977127	0.975493	0.977191	0.973682	0.982521

Fig. 14.

Comparison of adjusted -values for Polarizability.

Table 26.

Best regression models for PO based on adjusted values.

Predictor		Goodness of Fit				Significance		Regression Equation
Index	Model	r		Adj.	Std. Error	F-statistic	p-value
	Cubic	0.991222	0.982521	0.978151	1.888591	224.845079	8.30e-11
	Cubic	0.984650	0.969535	0.961919	2.493319	127.299070	2.31e-09
	Cubic	0.964761	0.930764	0.913455	3.758760	53.773428	3.13e-07
	Cubic	0.964495	0.930250	0.912813	3.772682	53.347838	3.27e-07
	Logarithmic	0.932271	0.869129	0.859781	4.784400	92.975338	1.46e-07
	Quadratic	0.927704	0.860636	0.839195	5.123580	40.140327	2.74e-06	(1e-06)
	Linear	0.913868	0.835155	0.823380	5.369623	70.928037	7.49e-07

The comparison of the actual and the predicted values of polarizability of drugs by using the best forecasted index is illustrated in Table 27. The related graphical comparison is displayed in Fig. 15.

Table 27.

Predicted vs Actual values for PO using the overall best model.

Sample	Actual Value	Predicted Value	Residual	Sample	Actual Value	Predicted Value	Residual
	61.5	61.11459	0.385409		42.6	43.04924	−0.449242
	46.1	45.28386	0.816143		41.8	41.88624	−0.086241
	47.6	49.61778	−2.017779		45.7	43.04924	2.650758
	17.7	17.51328	0.186725		15.5	20.00449	−4.504490
	33.9	33.75051	0.149487		17.7	14.86552	2.834481
	30.6	30.40022	0.199777		48.4	48.53066	−0.130656
	29.4	28.56838	0.831620		28.5	28.56838	−0.068380
	39.5	40.68182	−1.181815		38.5	38.11580	0.384203

Fig. 15.

Actual and predicted values for Polarizability.

Analysis of regression models for the molar volume

Table 28 provides the values and adjusted values of cubic, quadratic, logarithmic, and linear regression models by assuming the sets of values of eccentricity-based invariants as independent variables and the set of values of molar volume of sixteen drugs as the dependent variable. Figure 16 illustrates a graphical comparison of the adjusted values for these regression models. When we analyse the adjusted values given in Table 28, we see that provides best value within the quadratic model, gives best value within the linear model, and have the best values within the cubic model, while the other invariants achieve their most suitable fits in the logarithmic model. Table 29 displays the statistical analysis of the best regression models for each independent variable. This analysis possesses the p value, standard error, F-statistic, correlation coefficient r, , adjusted , and regression equation. This statistical examination shows that the eccentric Albertson index is the most predictive index utilizing a logarithmic model.

Table 28.

Comparison of and adjusted -values for different regression models for MV.

Topological	Linear		Logarithmic		Quadratic		Cubic
Index		Adjusted		Adjusted		Adjusted		Adjusted
	0.877078	0.868298	0.916443	0.910475	0.922256	0.910295	0.932683	0.915853
	0.746801	0.728716	0.863104	0.853326	0.874375	0.855048	0.876738	0.845923
	0.922835	0.917324	0.936960	0.932457	0.935595	0.925687	0.940750	0.925938
	0.813248	0.799908	0.816353	0.803235	0.816331	0.788074	0.824273	0.780341
	0.775586	0.759557	0.772828	0.756601	0.776494	0.742108	0.779033	0.723792
	0.876800	0.868000	0.915662	0.909637	0.921932	0.909922	0.932224	0.915280
	0.931461	0.926565	0.965298		0.960078	0.953936	0.969763	0.962203

Fig. 16.

Comparison of adjusted -values for molar volume.

Table 29.

Best regression models for MV based on adjusted values.

Predictor		Goodness of Fit				Significance		Regression Equation
Index	Model	r		Adj.	Std. Error	F-statistic	p-value
	Logarithmic	0.982496	0.965298	0.962819	17.538671	389.436981	1.29e-11
	Logarithmic	0.967967	0.936960	0.932457	23.638944	208.081783	8.53e-10
	Cubic	0.965755	0.932683	0.915853	26.385002	55.420125	2.65e-07	(1e-06)
	Cubic	0.965517	0.932224	0.915280	26.474750	55.017946	2.76e-07
	Quadratic	0.935080	0.874375	0.855048	34.629854	45.241353	1.39e-06	(8e-06)
	Logarithmic	0.903523	0.816353	0.803235	40.347106	62.233226	1.61e-06
	Linear	0.880674	0.775586	0.759557	44.601067	48.384771	6.70e-06

The comparison of the actual and the predicted values of molar volume of drugs by using the best forecasted index is illustrated in Table 30. The related graphical comparison is displayed in Fig. 17.

Table 30.

Predicted vs Actual values for MV using the overall best model.

Sample	Actual Value	Predicted Value	Residual	Sample	Actual Value	Predicted Value	Residual
	451.9	438.2576	13.642367		323.9	333.8990	−9.998976
	376.6	350.6824	25.917627		333.0	325.0521	7.947917
	351.9	381.1299	−29.229905		327.6	333.8990	−6.298976
	139.6	147.2544	−7.654367		135.9	165.2821	−29.382077
	244.3	263.9166	−19.616640		142.9	127.7830	15.117043
	258.7	239.5701	19.129939		364.0	373.8678	−9.867791
	236.7	226.4175	10.282480		239.7	226.4175	13.282480
	312.6	315.8713	−3.271266		306.4	296.3999	10.000144

Fig. 17.

Actual and predicted values for molar volume.

Validation

To validate the performance of our regression models, we considered five coronary artery disease drug structures: valsartan, sacubitril, rivaroxaban, perindopril, and ticagrelor. The molecular graphs of these drug structures are displayed in Fig. 18.

Fig. 18.

Chemical structures of the studied compounds: (a) Valsartan, (b) Sacubitril, (c) Rivaroxaban, (d) Perindopril, and (e) Ticagrelor.

The experimental values for physicochemical properties of these compounds were acquired from ChemSpider and PubChem and are outlined in Table 31. By using eccentricity-based edge-partitioning procedures, we computed the eccentricity-based invariants for drug structures (see Table 32) and used our regression models to forecast their properties. The predictive performance was estimated using MAE and RMSE. A comparison between the experimentally model-predicted and observed values (shown in Table 33) showed close alignment. These outcomes indicate that the linear, logarithmic, cubic, and quadratic regression models designed in this study act as reliable predictive means for assessing key physicochemical properties of coronary artery disease drug structures.

Table 31.

Physico-chemical properties of drugs.

Drugs	MW (g/mol)	HAC	CO	BP	ENV (kj/mol)	MR	PO	MV
Valsartan	435.5	32	608	684.9	105.5	120.6	47.8	359.1
Sacubitril	411.5	30	550	656.9	101.6	113.6	45	357.4
Rivaroxaban	435.9	29	645	732.6	106.9	106.7	42.3	298.5
Perindopril	368.5	26	524	537.4	89.1	96.2	38.1	320.4
Ticagrelor	522.6	36	736	777.6	118.7	126.3	50.1	311.9

Table 32.

Topological Indices of Drug Molecules.

Drug molecules
Valsartan	793	4749	33.969953	13.601934	7.123505	197.913319	31
Sacubitril	720	4298	30.970276	12.427407	6.508990	179.671143	30
Rivaroxaban	822	5474	31.977398	12.286244	6.410039	205.223735	28
Perindopril	500	2368	26.958165	11.963250	6.341157	124.637342	26
Ticagrelor	1041	6982	39.969904	15.230152	7.936053	259.883118	37

Table 33.

Experimental and Predicted Values with Error Metrics.

Property	Drugs
	Valsartan	Sacubitril	Rivaroxaban	Perindopril	Ticagrelor
Molecular weight
Experimental	435.5	411.5	435.9	368.5	522.6
Predicted	453.161382	418.3792	430.1606	370.4077	519.9526
RMSE	17.66	6.88	5.74	1.91	2.65
MAE	17.66	6.88	5.74	1.91	2.65
Heavy atom count
Experimental	32	30	29	26	36
Predicted	32.160255	29.4646	30.3696	25.8591	37.5521
RMSE	0.16	0.54	1.37	0.14	1.55
MAE	0.16	0.54	1.37	0.14	1.55
Complexity
Experimental	608	550	645	524	736
Predicted	652.887171	551.5214	538.9274	510.0584	768.0438
RMSE	44.89	1.52	106.07	13.94	32.04
MAE	44.89	1.52	106.07	13.94	32.04
Boiling point
Experimental	684.9	656.9	732.6	537.4	777.6
Predicted	637.133657	603.2051	614.9496	552.2869	696.8263
RMSE	47.77	53.69	117.65	14.89	80.77
MAE	47.77	53.69	117.65	14.89	80.77
Enthalpy of vaporization
Experimental	105.5	101.6	106.9	89.1	118.7
Predicted	99.544922	97.1599	92.0856	86.7628	110.3402
RMSE	5.96	4.44	14.81	2.34	8.36
MAE	5.96	4.44	14.81	2.34	8.36
Molar Refractivity
Experimental	120.6	113.6	106.7	96.2	126.3
Predicted	116.954328	114.2073	108.5708	102.6014	133.7173
RMSE	3.65	0.61	1.87	6.40	7.42
MAE	3.65	0.61	1.87	6.40	7.42
Polarizability
Experimental	47.8	45.0	42.3	38.1	50.1
Predicted	46.359014	45.2839	43.0492	40.6818	53.0088
RMSE	1.44	0.28	0.75	2.58	2.91
MAE	1.44	0.28	0.75	2.58	2.91
Molar volume
Experimental	359.1	357.4	298.5	320.4	311.9
Predicted	358.658917	350.6824	333.8990	315.8713	401.6996
RMSE	0.44	6.72	35.40	4.53	89.80
MAE	0.44	6.72	35.40	4.53	89.80

Conclusion

Contributions

This study utilized the eccentricity-based edge partitioning approaches to compute the associated graph invariants of drugs used to treat coronary artery disease. In this examination, the cubic, quadratic, linear, and logarithmic models were developed, and their prediction performance was compared. The comparison showed that most of the physicochemical properties were best predicted using the cubic models. Quadratic regression analysis revealed the eccentric geometric arithmetic index as the top predictor of molecular weight, with the eccentric Albertson index a close second. The eccentric atom bond connectivity and eccentric sum connectivity indices also showed strong predictive ability in linear regression, ranking third and fourth, respectively. For the heavy atom count, linear regression analysis showed that the eccentric geometric arithmetic index was the best predictor, followed by the eccentric Albertson index. The eccentric atom bond connectivity index in linear regression, and eccentric first Zagreb index in quadratic regression, were third and fourth, respectively. For complexity, the eccentric atom bond connectivity index was the best predictor in cubic regression, followed by the eccentric sum connectivity index. The eccentric geometric arithmetic index and eccentric Albertson index were third and fourth, respectively. For the boiling point, logarithmic regression analysis showed that the eccentric geometric arithmetic index was the best forecaster, followed closely by the eccentric Albertson index in cubic regression. The eccentric first Zagreb index and eccentric inverse sum index in quadratic regression were third and fourth, respectively. For the enthalpy of vaporization, cubic regression analysis showed that the eccentric Albertson index was the top predictor, followed by the eccentric geometric arithmetic index in linear regression. The eccentric first Zagreb index and eccentric inverse sum index in cubic regression were third and fourth, respectively. For molar refractivity, and polarizability, cubic regression analysis showed that the eccentric Albertson index was the top predictor, followed by the eccentric geometric arithmetic index. The eccentric first Zagreb index and eccentric inverse sum index were third and fourth, respectively. For molar volume, logarithmic regression analysis showed that the eccentric Albertson index was the top predictor, followed closely by the eccentric geometric arithmetic index. The eccentric first Zagreb index and eccentric inverse sum index in cubic regression were third and fourth, respectively. These outcomes improve the development of drugs for coronary artery disease by helping the forecast of physicochemical properties, thereby helping more efficient optimization and identification of potential therapeutics.

Study implications

• i.Improved estimation exactness for molar refractivity, boiling points, and other fundamental physicochemical properties.
• ii.Suggested scalable tools for molecular screening and drug development.
• iii.Provided the cheminformatics applications in pharmaceuticals and material science.

Limitations

• i.The emphasis on eccentricity-based indices may overlook quantum chemical and steric effects, necessitating hybrid models for more broader applicability.
• ii.This analysis focuses solely on drugs that treat coronary artery disease, limiting generalizability to other classes of drugs.
• iii.Standard environmental prerequisites and simple regression models may not completely catch real-world molecular behavior.
• iv.A small dataset can increase the risk of overfitting in regression models, which makes it essential to utilize regularization strategies.

Future study

• i.Extend datasets to contain various molecular graphs.
• ii.Analyze further properties like pharmacokinetics, solubility, and toxicity.
• iii.Integrate machine learning for improved predict performance.
• iv.Examine real-world applications in advanced material design and drug discovery.

Author contributions

Conceptualization: NI, SA, AMA, ZI, AA, KAT; Methodology: NI, SA, ZI; Formal Analysis: SA, AMA, ZI, AA, KAT; Validation: NI, SA,AMA, ZI, AA, KAT; Writing original draft:SA, ZI, KAT; Writing review and editing: NI, SA, AMA, ZI, AA, KAT.

Data availability

All data generated or analysed during this study are included in this published article.

Declarations

Competing interests

The authors declare no competing interests.