The study begins with a deceptively simple transformation. Each of the fifteen drug molecules, ranging from the classic estrogen-blocker tamoxifen to the PARP inhibitor olaparib, was converted into a molecular graph, a network in which atoms become vertices and chemical bonds become edges. Once a drug is a graph, the full arsenal of chemical graph theory becomes available. The researchers focused on so-called resolving sets, a concept first introduced by Peter Slater in 1975 and independently explored by Frank Harary and Robert Melter. A resolving set is a small collection of vertices whose distances to every other vertex in the network are unique, allowing every atom to be unambiguously identified by its pattern of distances alone. The minimum size of such a set is called the metric dimension, and it acts as a fingerprint of structural complexity.

Finding these sets is computationally hard, so the team wrote a brute-force algorithm that exhaustively tests vertex combinations until it identifies all minimal resolving sets. The approach is efficient for the molecules considered here, which have fewer than sixty carbon atoms, but the authors acknowledge that it becomes prohibitively slow as molecular size grows. For each drug, the algorithm returned a metric dimension, with fulvestrant and anastrozole scoring five, tamoxifen, toremifene, megestrol acetate, alpelisib, and ribociclib scoring four, and smaller, less branched molecules such as elacestrant scoring just two.

From these resolving sets, the researchers derived a new family of descriptors they call resolving topological indices, computed via a two-variable polynomial that generalizes the well-known M-polynomial framework of Deutsch and Klavžar. Traditional degree-based indices, such as the Randić and Zagreb indices, summarize branching and connectivity but lose information about how vertices are positioned relative to one another. Resolving indices add a second layer, weighting each vertex by its contribution to uniquely identifying the rest of the molecule. In principle, this captures structural nuances that conventional descriptors overlook, which is precisely why the authors believe these indices could sharpen future Quantitative Structure–Property Relationship models, the statistical machinery that links molecular architecture to biological behavior.

The next step was to connect the mathematics to pharmacology. Using data drawn from the pkCSM database, the team assembled five ADMET properties for each drug: water solubility, Caco-2 permeability, skin permeability, oral acute toxicity, and oral chronic toxicity. These parameters describe whether a drug dissolves, crosses the gut wall, penetrates membranes, and remains safe in the short and long term. Quadratic regression models then tested how well the resolving indices predicted each property. The results were sobering but informative. Water solubility, acute toxicity, and chronic toxicity showed weak-to-moderate correlations, with chronic toxicity performing best: one index achieved a correlation coefficient of 0.670, and the strongest regression model for that property explained roughly 45 percent of the variance with statistical significance. Caco-2 permeability and skin permeability, by contrast, correlated poorly, with coefficients often below 0.55, a reminder that topology alone cannot capture every pharmacokinetic subtlety.

The real innovation came when the researchers stopped trying to predict individual properties and instead asked which drug performs best across all of them simultaneously. This is the domain of multi-criteria decision-making, or MCDM, a family of operations-research techniques usually applied to engineering and energy planning. To avoid subjective judgment, the team used the entropy method, which assigns weights to each criterion based purely on how much the data vary across drugs; criteria with greater variability carry more information and receive larger weights. The resulting weights hovered around 0.15 to 0.21 for each of the five properties, with water solubility, skin permeability, and both toxicity measures weighted slightly above Caco-2 permeability.

Two ranking engines were then set to work. The first, TOPSIS, the Technique for Order Preference by Similarity to Ideal Solution, positions each drug in a normalized multidimensional space and measures its Euclidean distance to an idealized best-case drug and a worst-case drug; the winner is the molecule geometrically closest to ideal. The second, Simple Additive Weighting, or SAW, is even more direct: each drug’s normalized, weighted scores are simply summed, and the largest total wins. Because the two methods rest on different mathematical logic, agreement between them is a useful robustness check, and a comparison plot showed strong consistency across the fifteen drugs.

The verdicts were clear. Fulvestrant, the estrogen-receptor degrader, earned the number-one rank under both methods, followed by neratinib, an irreversible HER-family kinase inhibitor, and ribociclib, a CDK4/6 inhibitor. Toremifene, megestrol acetate, anastrozole, and alpelisib occupied the middle of the table, while letrozole and exemestane, two aromatase inhibitors, fell to the bottom. Notably, the ranking partially echoes clinical reality: ribociclib combined with fulvestrant has been shown in the MONALEESA-3 trial program to improve overall survival in advanced hormone-receptor-positive, HER2-negative breast cancer, a convergence the authors cite as evidence that the method has heuristic value.

Yet the researchers are emphatic that the league table is a mathematical artifact, not a clinical prescription. The ranking reflects only computed topological indices and predicted ADMET values, with no experimental or clinical data feeding the model. The entropy weighting scheme is blind to biology: it cannot know that toxicity matters more than skin permeability in real-world oncology, regardless of statistical variability. The study also covers only fifteen approved drugs and a handful of ADMET endpoints, and the brute-force resolving-set algorithm limits scalability to larger molecular systems. The authors recommend that future iterations incorporate expert judgment and biologically informed weighting, and suggest extending the framework to larger datasets, entropy-based topological indices, and additional physicochemical properties.

Even with those caveats, the study sketches a provocative vision of drug evaluation in which molecules are judged as networks, properties are forecast by polynomials, and treatment candidates are sorted by distance from a mathematical ideal. As computational screening becomes ever more central to drug discovery, frameworks like this one, blending chemical graph theory, regression modeling, and decision science, may help researchers triage vast candidate libraries before a single experiment is run. For breast cancer, a disease whose therapies already span hormones, kinases, microtubules, and DNA repair, having a common mathematical language to compare such disparate drugs is a genuinely novel contribution, even if the final arbiter remains the clinic.

Subject of Research: Computational ranking of breast cancer therapeutics using resolving topological indices, ADMET analysis, and multi-criteria decision-making methods

Article Title: Integration of resolving topological indices and ADMET analysis with MCDM methods (TOPSIS and SAW) for ranking breast cancer therapeutics

Article References: Integration of resolving topological indices and ADMET analysis with MCDM methods (TOPSIS and SAW) for ranking breast cancer therapeutics. (n.d.). https://doi.org/10.3389/fbinf.2026.1913227

Image Credits: AI Generated

DOI: 10.3389/fbinf.2026.1913227

Keywords: breast cancer, topological indices, chemical graph theory, ADMET, QSPR, TOPSIS, SAW, multi-criteria decision-making, fulvestrant, neratinib, ribociclib, metric dimension