QSPR Modeling of Degree-Based Topological Indices with Hepatocellular Carcinoma Drugs

Abstract

The study discussed the QSPR analysis of the mentioned topological indices. The study also demonstrated that the characteristics obtained are highly correlated with those of Hepatocellular Carcinoma (Liver Cancer) drugs through linear regression. drugs are represented as molecular graphs where each vertex represents an atom and each edge represents a link between two atoms. Consider G (V, E) as a molecular graph, where V is the set of vertices and E is the set of edges. In this study, used 6 degree based topological indices, M1(G), M2(G), F(G), S(G), Y(G) and D(G). These indices were used to model five representative physical properties of five liver cancer drugs: BP, FP, P, ST, and MV. The values for these properties were obtained from ChemSpider. The study concluded that degree-based topological indices are effective molecular descriptors for predicting the physical properties of liver cancer drugs. The regression models revealed significant correlations between Surface Tension (ST) and indices such as the Forgotten Index (F(G)) and the Sum-Connectivity Index (S(G)). Although other properties, such as Boiling Point (BP) and Flash Point (FP), demonstrated weaker correlations, the overall findings suggest that topological indices can be valuable tools in Quantitative Structure-Property Relationship (QSPR) studies.

Introduction

Hepatocellular Carcinoma (HCC) is a severe and widespread form of liver cancer, known for its high mortality rate due to late diagnosis and limited treatment options. Major causes include hepatitis B/C infections, alcohol abuse, obesity, and aflatoxin exposure. Symptoms include abdominal pain, weight loss, jaundice, and fatigue. Treatment typically involves surgery, chemotherapy, targeted therapy, immunotherapy, and radiation. Drug-based treatments are especially vital in managing disease progression.

Chemical Graph Theory in Drug Analysis

Chemical graph theory is a mathematical approach in which molecular structures are represented as graphs:

• Vertices represent atoms

• Edges represent chemical bonds

This method allows researchers to derive topological indices—numerical values that describe molecular structure and can predict chemical and biological properties. These indices are crucial in QSPR (Quantitative Structure-Property Relationship) and QSAR (Quantitative Structure-Activity Relationship) modeling, especially in pharmaceutical research.

Degree-Based Topological Indices Used:

These indices are calculated based on the degrees of atoms in the molecular graph. They help correlate structure with physical/chemical properties.

1. First Zagreb Index (M1): Sum of degrees of bonded atoms.

2. Second Zagreb Index (M2): Product of degrees of bonded atoms.

3. Forgotten Index (F): Sum of squares of degrees of bonded atoms.

4. Sum-Connectivity Index (S): Fourth powers of degrees.

5. Yemen Index (Y): Sum of cubes of degrees.

6. D Index (D): Sum of sixth powers of degrees.

Methodology

• Molecular structures of five liver cancer drugs were modeled as simple molecular graphs.

• Topological indices were calculated.

• Physical properties analyzed:

  • Boiling Point (BP)

  • Flash Point (FP)

  • Polarizability (P)

  • Surface Tension (ST)

  • Molar Volume (MV)

• Experimental data were obtained from ChemSpider.

• Linear regression was used to study the correlation between the topological indices and drug properties.

Findings

• The study demonstrated a strong correlation between the degree-based topological indices and the physicochemical properties of liver cancer drugs.

• This confirms the effectiveness of topological indices in QSPR modeling for HCC drug analysis.

• These insights can help in predicting drug behavior and potentially designing better therapeutic compounds for liver cancer.

Conclusion

This study showed that degree-based topological indices are effective molecular descriptors for predicting the physical properties of liver cancer drugs. The regression models revealed significant correlations between Surface Tension (ST) and indices such as the Forgotten Index (F(G)) and the Sum-Connectivity Index (S(G)). Although other properties, such as Boiling Point (BP) and Flash Point (FP), demonstrated weaker correlations, the overall findings suggest that topological indices can be valuable tools in Quantitative Structure-Property Relationship (QSPR) studies.

References

[1] Aslam, A., Ahmad, S., & Gao, W. (2017). On Certain Topological Indices of Boron Triangular Nanotubes. Zeitschrift Fur Naturforschung - Section A Journal of Physical Sciences, 72(8). https://doi.org/10.1515/zna-2017-0135 [2] Aslam, A., Bashir, Y., Ahmad, S., & Gao, W. (2017). On topological indices of certain dendrimer structures. Zeitschrift Fur Naturforschung - Section A Journal of Physical Sciences, 72(6). https://doi.org/10.1515/zna-2017-0081 [3] Calderon-Martinez, E., Landazuri-Navas, S., Vilchez, E., Cantu-Hernandez, R., Mosquera-Moscoso, J., Encalada, S., Al lami, Z., Zevallos-Delgado, C., & Cinicola, J. (2023). Prognostic Scores and Survival Rates by Etiology of Hepatocellular Carcinoma: A Review. In Journal of Clinical Medicine Research (Vol. 15, Issue 4). https://doi.org/10.14740/jocmr4902 [4] Chaudhry, F., Husin, M. N., Afzal, F., Afzal, D., Ehsan, M., Cancan, M., & Farahani, M. R. (2021). M-polynomials and degree-based topological indices of tadpole graph. Journal of Discrete Mathematical Sciences and Cryptography, 24(7). https://doi.org/10.1080/09720529.2021.1984561 [5] Costa, P. C. S., Evangelista, J. S., Leal, I., & Miranda, P. C. M. L. (2021). Chemical graph theory for property modeling in qsar and qspr—charming QSAR & QSPR. Mathematics, 9(1). https://doi.org/10.3390/math9010060 [6] Elahi, K., Ahmad, A., & Hasni, R. (2018). Construction algorithm for zero divisor graphs of finite commutative rings and their vertex-based eccentric topological indices. Mathematics, 6(12). https://doi.org/10.3390/math6120301 [7] Figuerola, B., & Avila, C. (2019). The phylum bryozoa as a promising source of anticancer drugs. In Marine Drugs (Vol. 17, Issue 8). https://doi.org/10.3390/md17080477 [8] Furtula, B., & Gutman, I. (2015). A forgotten topological index. Journal of Mathematical Chemistry, 53(4). https://doi.org/10.1007/s10910-015-0480-z [9] Gao, W., Wang, W., & Farahani, M. R. (2016). Topological indices study of molecular structure in anticancer drugs. Journal of Chemistry, 2016. https://doi.org/10.1155/2016/3216327 [10] Govardhan, S., Roy, S., Prabhu, S., & Siddiqui, M. K. (2023). Computation of Neighborhood M-Polynomial of Three Classes of Polycyclic Aromatic Hydrocarbons. Polycyclic Aromatic Compounds, 43(6). https://doi.org/10.1080/10406638.2022.2103576 [11] Gutman, I., & Trinajsti?, N. (1972). Graph theory and molecular orbitals. Total ?-electron energy of alternant hydrocarbons. Chemical Physics Letters, 17(4). https://doi.org/10.1016/0009-2614(72)85099-1 [12] Hayat, S., Imran, M., & Liu, J. B. (2019). Correlation between the Estrada index and ?-electronic energies for benzenoid hydrocarbons with applications to boron nanotubes. International Journal of Quantum Chemistry, 119(23). https://doi.org/10.1002/qua.26016 [13] Herrero, R., Carvajal, L. J., Camargo, M. C., Riquelme, A., Porras, C., Ortiz, A. P., Camargo, L. A., Fink, V., van De Wyngard, V., Lazcano-Ponce, E., Canelo-Aybar, C., Balbin-Ramon, G., Feliu, A., & Espina, C. (2023). Latin American and the Caribbean Code Against Cancer 1st edition: Infections and cancer. Cancer Epidemiology, 86. https://doi.org/10.1016/j.canep.2023.102435 [14] Hosamani, S., Perigidad, D., Jamagoud, S., Maled, Y., & Gavade, S. (2017). QSPR Analysis of Certain Degree Based Topological Indices. Journal of Statistics Applications & Probability, 6(2). https://doi.org/10.18576/jsap/060211 [15] Ismael, M., Zaman, S., Elahi, K., Koam, A. N. A., & Bashir, A. (2024). Analytical Expressions and Structural Characterization of Some Molecular Models Through Degree Based Topological Indices. Mathematical Modelling of Engineering Problems, 11(1). https://doi.org/10.18280/mmep.110105 [16] Kavitha, S. R. J., Abraham, J., Arockiaraj, M., Jency, J., & Balasubramanian, K. (2021). Topological Characterization and Graph Entropies of Tessellations of Kekulene Structures: Existence of Isentropic Structures and Applications to Thermochemistry, Nuclear Magnetic Resonance, and Electron Spin Resonance. Journal of Physical Chemistry A, 125(36). https://doi.org/10.1021/acs.jpca.1c06264 [17] Kotsari, M., Dimopoulou, V., Koskinas, J., & Armakolas, A. (2023). Immune System and Hepatocellular Carcinoma (HCC): New Insights into HCC Progression. In International Journal of Molecular Sciences (Vol. 24, Issue 14). https://doi.org/10.3390/ijms241411471 [18] Kumar, S. (2015). Drug Targets for Cancer Treatment: An Overview. Medicinal Chemistry, 5(3). https://doi.org/10.4172/2161-0444.1000252 [19] Kwun, Y. C., Ali, A., Nazeer, W., Ahmad Chaudhary, M., & Kang, S. M. (2018). M-polynomials and degree-based topological indices of triangular, hourglass, and jagged-rectangle benzenoid systems. Journal of Chemistry, 2018. https://doi.org/10.1155/2018/8213950 [20] M.C., S., N.S., B., & K.N., A. (2020). Predicting physico-chemical properties of octane isomers using QSPR approach. Malaya Journal of Matematik, 8(1). https://doi.org/10.26637/mjm0801/0018 [21] Nagarajan, S., & Durga, M. (2023). Computing Y-index of Different Corona Products of Graphs. Asian Research Journal of Mathematics, 19(10). https://doi.org/10.9734/arjom/2023/v19i10729 [22] Nagarajan, S., & Durga, M. (2024). A Computational Approach on Fenofibrate Drug Using Degree-Based Topological Indices and M-polynomials. Asian Journal of Chemical Sciences, 14(2). https://doi.org/10.9734/ajocs/2024/v14i2293 [23] Nandi, S., & C. Bagchi, M. (2012). Importance of Kier-Hall Topological Indices in the QSAR of Anticancer Drug Design. Current Computer Aided-Drug Design, 8(2). https://doi.org/10.2174/157340912800492384 [24] Shaheen, N. J., Hansen, R. A., Morgan, D. R., Gangarosa, L. M., Ringel, Y., Thiny, M. T., Russo, M. W., & Sandler, R. S. (2006). The burden of gastrointestinal and liver diseases, 2006. American Journal of Gastroenterology, 101(9). https://doi.org/10.1111/j.1572-0241.2006.00723.x [25] Sharma, M., & Parveen, R. (2021). The Application of Image Processing in Liver Cancer Detection. International Journal of Advanced Computer Science and Applications, 12(10). https://doi.org/10.14569/IJACSA.2021.0121050 [26] Singh, M., Sharma, S. K., & Bhat, V. K. (2023). Vertex-Based Resolvability Parameters for Identification of Certain Chemical Structures. ACS Omega, 8(42). https://doi.org/10.1021/acsomega.3c06306 [27] Zhang, C., & Yang, M. (2021). The emerging factors and treatment options for nafld-related hepatocellular carcinoma. In Cancers (Vol. 13, Issue 15). https://doi.org/10.3390/cancers13153740 [28] Zhou, B., & Trinajsti?, N. (2010). On general sum-connectivity index. Journal of Mathematical Chemistry, 47(1). https://doi.org/10.1007/s10910-009-9542-4

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Copyright © 2025 S. Ranjitham, O. V. Shanmuga Sundaram, G. Priyadharsini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.