Mathematical Modeling of H1-Antihistamines: A QSPR Approach Using Topological Indices

Abstract

Allergic diseases represent a significant global health burden, requiring effective and safe therapeutic agents for long-term management. H1-antihistamines are among the most widely prescribed and over-the-counter drugs for treating allergic conditions, yet their variable physicochemical and pharmacokinetic properties present challenges in optimizing drug selection, safety, and efficacy. A systematic exploration of their structure–property relationships is, therefore, essential for guiding rational drug design. In this study, the Quantitative Structure–Property Relationship (QSPR) of a selection of H1-antihistamines, including both conventional and second-generation compounds, is investigated by using degree-based topological indices and linear regression models. The computed indices are systematically correlated to key physicochemical properties, revealing strong and statistically significant relationships. These findings provide deeper insights into the molecular factors influencing drug behavior and highlight the predictive utility of topological descriptors. Overall, the developed QSPR models not only enhance the understanding of H1-antihistamines but also establish a framework that can accelerate the identification and optimization of next-generation agents with improved pharmacological profiles.

1. Introduction

Histamine, an endogenous amine found throughout animal tissues, is involved in various physiological processes, most notably in immune responses and allergic reactions. Its effects are mediated through histamine receptors, of which H1 and H2 are the most pharmacologically significant. The therapeutic management of histamine-mediated conditions is primarily achieved using antihistamines, which are the agents that block the action of histamines at these receptor sites.

Antihistamines targeting H1-receptors are commonly used in the treatment of allergic conditions, such as rhinitis, urticaria, and anaphylaxis. These agents are broadly classified into first- and second-generation compounds based on their ability to cross the blood–brain barrier and their associated side effect profiles. First-generation H1-antihistamines tend to interact with both central and peripheral receptors, often leading to sedative effects. In contrast, second-generation compounds are more selective for peripheral receptors and are associated with a lower incidence of central nervous system adverse effects. , Refer to Table for the classification of drugs used in this study.

1. List of H1-Antihistamines Used in the Study.

classification	name
first generation/highly sedative	diphenhydramine
	promethazine
	hydroxyzine
first generation/moderately sedative	pheniramine
	cyproheptadine
	meclizine
	cinnarizine
first generation/mild sedative	chlorpheniramine
	triprolidine
	clemastine
second generation	fexofenadine
	loratadine
	desloaratadine
	cetirizine
	azelastine
	ebastine
	rupatadine

Although both first- and second-generation antihistamines are widely available, often as over-the-counter medications, their diverse physicochemical and pharmacokinetic properties continue to pose challenges in terms of drug selection, safety, and therapeutic efficacy. The extensive clinical use of these agents, coupled with the demand for improved therapeutic profiles, highlights the necessity of rational drug design to develop newer and more selective compounds. Moreover, recent studies have suggested additional clinical benefits beyond allergy treatment. For instance, desloratadine and loratadine have been associated with substantially improved survival outcomes in breast cancer and melanoma patients, with desloratadine showing benefits across immunogenic tumors and loratadine improving survival in selected tumor types. In addition, H1-antihistamines find broad applications in dermatology, further underscoring their therapeutic relevance.

In recent years, computational approaches have become integral to pharmaceutical research, particularly in the early stages of drug development. Among these, quantitative structure–property relationship (QSPR) modeling has proven to be a powerful tool for predicting the physicochemical and biological properties of compounds based on their molecular structure. QSPR methods enhance the efficiency of drug discovery by providing a cost-effective means of screening and optimizing candidate molecules.

Central to QSPR modeling is the use of molecular descriptors, with topological indices (TIs) playing a particularly significant role. These indices are derived from graph-theoretical representations of molecules, where atoms are treated as vertices and chemical bonds are treated as edges. Degree-based topological indices (DTIs) capture essential aspects of molecular connectivity and branching, offering valuable insights into the structural attributes that influence the chemical behavior and biological activity. Beyond the conventional degree-based indices, TIs derived from other graph parameters have gained significant attention in recent years. Among these, neighborhood degree-based topological indices (NDTIs) are particularly noteworthy. Unlike traditional indices that account for only the immediate degree of a vertex, NDTIs incorporate information from the extended neighborhood, thereby capturing an additional layer of structural branching. This makes the comparison between the conventional and modified forms of TIs both relevant and valuable.

QSPR analyses employing DTIs have been widely applied to various therapeutic classes of drugs, including those used in the treatment of cancer, hepatitis, malaria, breast cancer, pyelonephritis, Lyme disease, glaucoma, postpartum depression, and as antibiotics. In addition, NDTIs have been effectively employed in the analysis of various drug properties, including those of asthma medications, antituberculosis drugs, heart transplant drugs, and anesthetic agents. There remains a significant gap in the literature concerning the analysis of diverse properties of H1-antihistamines in relation to their chemical structures. The present study addresses this gap by providing valuable insights into the structural determinants of drug behavior and by demonstrating the predictive potential of topological descriptors.

The present study applies 12 DTIs and ten NDTIs to a selection of H1-antihistamines in order to model and predict key physicochemical properties. The chemical structures of the set of H1-antihistamines analyzed in this study are listed in Figure . These structures were retrieved from the PubChem database. Regression analysis is employed to establish relationships between the indices and these properties, enabling the development of predictive models. These models are further used to evaluate additional antihistamines, including cyclizine and doxylamine, with the aim of assessing their characteristics within the same analytical framework. This methodological approach contributes to a deeper understanding of the structural basis for drug behavior and supports the broader goal of designing next-generation antihistamines with enhanced efficacy and safety profiles.

1.

Chemical structures of H1-antihistaminics.

2. Computation of Topological Indices

In this study, all graphs G = (V, E) are assumed to be connected, where the vertex set V(G) and edge set E(G) represent the atoms and chemical bonds of a molecule, respectively. Each vertex v ∈ V(G) has a degree d _v , indicating the number of vertices adjacent to it, an idea analogous to the chemical concept of valence, which describes the bonding capacity of an atom. The notation N(v) denotes the set of vertices adjacent to v in G.

The chemical graphs analyzed here were constructed from the molecular structures of selected H1-antihistamines. For simplicity, distinctions between single and double bonds are not considered, and hydrogen atoms are omitted from the graph structure, following standard practice in chemical graph theory.

2.1. Degree-Based Topological Indices

The topological indices defined in Table are based on the degrees of the vertices in the given graph. These indices are calculated using the edge partition technique, which categorizes edges according to the degrees of their end points. The quantities $E_{d_u,d_v}$ denote the number of edges connecting vertices of degrees d _u and d _v , with their corresponding values presented in Table .

2. List of DTIs Used in the Study.

atom-bond connectivity index	$$ABC(G)=\sum _{z_1{z}_2\in E(G)}\sqrt{\frac{d_{z_1}+d_{z_2}-2}{d_{z_1}\times d_{z_2}}}$$
atom-bond sum-connectivity index	$$ABS(G)=\sum _{z_1{z}_2\in E(G)}\sqrt{\frac{d_{z_1}+d_{z_2}-2}{d_{z_1}+d_{z_2}}}$$
forgotten index	$$F(G)=\sum _{z_1{z}_2\in E(G)}[{d_{z_1}}^2+{d_{z_2}}^2]$$
geometric–arithmetic index	$$GA(G)=\sum _{z_1{z}_2\in E(G)}\frac{2\sqrt{d_{z_1}\times d_{z_2}}}{d_{z_1}+d_{z_2}}$$
harmonic index	$$H(G)=\sum _{z_1{z}_2\in E(G)}\frac{2}{d_{z_1}+d_{z_2}}$$
inverse sum index	$$IS(G)=\sum _{z_1{z}_2\in E(G)}\frac{d_{z_1}\times d_{z_2}}{d_{z_1}+d_{z_2}}$$
Randić index	$$R(G)=\sum _{z_1{z}_2\in E(G)}\frac{1}{\sqrt{d_{z_1}\times d_{z_2}}}$$
sum-connectivity index	$$SCI(G)=\sum _{z_1{z}_2\in E(G)}\frac{1}{\sqrt{d_{z_1}+d_{z_2}}}$$
symmetric division index	$$SD(G)=\sum _{z_1{z}_2\in E(G)}\frac{d_{z_1}}{d_{z_2}}+\frac{d_{z_2}}{d_{z_1}}$$
first Zagreb index	$$M_1(G)=\sum _{z_1{z}_2\in E(G)}[d_{z_1}+d_{z_2}]$$
hyper-Zagreb index	$$HM(G)=\sum _{z_1{z}_2\in E(G)}{[d_{z_1}+d_{z_2}]}^2$$
second Zagreb index	$$M_2(G)=\sum _{z_1{z}_2\in E(G)}[d_{z_1}\times d_{z_2}]$$

3. Edge Partition of the Candidate Drugs in Terms of Vertex Degrees.

sl. no.	name	E 1,2	E 1,3	E 1,4	E 2,2	E 2,3	E 2,4	E 3,3	E 3,4
1	diphenhydramine	0	2	0	10	6	0	2	0
2	promethazine	0	3	0	6	8	0	5	0
3	hydroxyzine	1	1	0	12	11	0	3	0
4	pheniramine	0	2	0	9	6	0	2	0
5	cyproheptadine	0	1	0	9	10	0	5	0
6	meclizine	0	2	0	10	16	0	3	0
7	cinnarizine	0	0	0	16	12	0	3	0
8	chlorpheniramine	0	3	0	7	8	0	2	0
9	triprolidine	0	1	0	10	10	0	2	0
10	clemastine	0	2	1	10	9	1	1	2
11	fexofenadine	0	3	3	14	14	0	1	5
12	loratadine	1	2	0	8	13	0	6	0
13	desloaratadine	0	1	0	9	10	0	5	0
14	cetirizine	0	3	0	11	12	0	3	0
15	azelastine	0	3	0	8	14	0	5	0
16	ebastine	0	1	3	14	16	0	3	1
17	rupatadine	0	2	0	9	18	0	5	0

As an example, Theorem 1 presents the detailed computation of the DTIs for diphenhydramine.

Theorem 1. For the chemical graph C of diphenhydramine, M ₁(C) = 90, M ₂(C) = 100, H(C) = 9.0667, HM(C) = 414, ABC(C) = 14.28, R(C) = 9.2709, GA(C) = 19.6108, SCI(C) = 9.4998, IS(C) = 21.7, SD(C) = 43.6667, F(C) = 214, and ABS(C) = 14.7659.

Proof. Using the definitions provided in Table and the edge partitions detailed in Table

$$M_1(C)=E_{1,3}(1+3)+E_{2,2}(2+2)+E_{2,3}(2+3)+E_{3,3}(3+3)=90$$ 1a

$$M_2(C)=E_{1,3}(1\times 3)+E_{2,2}(2\times 2)+E_{2,3}(2\times 3)+E_{3,3}(3\times 3)=100$$ 1b

$$H(C)=E_{1,3}\left(\frac{2}{1+3}\right)+E_{2,2}\left(\frac{2}{2+2}\right)+E_{2,3}\left(\frac{2}{2+3}\right)+E_{3,3}\left(\frac{2}{3+3}\right)=9.0667$$ 1c

$$HM(C)=E_{1,3}{(1+3)}^2+E_{2,2}{(2+2)}^2+E_{2,3}{(2+3)}^2+E_{3,3}{(3+3)}^2=414$$ 1d

$$ABC(C)=E_{1,3}\sqrt{\frac{1+3-2}{1\times 3}}+E_{2,2}\sqrt{\frac{2+2-2}{2\times 2}}+E_{2,3}\sqrt{\frac{2+3-2}{2\times 3}}+E_{3,3}\sqrt{\frac{3+3-2}{3\times 3}}=14.28$$ 1e

$$R(C)=\frac{E_{1,3}}{\sqrt{1\times 3}}+\frac{E_{2,2}}{\sqrt{2\times 2}}+\frac{E_{2,3}}{\sqrt{2\times 3}}+\frac{E_{3,3}}{\sqrt{3\times 3}}=9.2709$$ 1f

$$GA(C)=2E_{1,3}\left(\frac{\sqrt{1\times 3}}{1+3}\right)+2E_{2,2}\left(\frac{\sqrt{2\times 2}}{2+2}\right)+2E_{2,3}\left(\frac{\sqrt{2\times 3}}{2+3}\right)+2E_{3,3}\left(\frac{\sqrt{3\times 3}}{3+3}\right)=19.6108$$ 1g

$$SCI(C)=\frac{E_{1,3}}{\sqrt{1+3}}+\frac{E_{2,2}}{\sqrt{2+2}}+\frac{E_{2,3}}{\sqrt{2+3}}+\frac{E_{3,3}}{\sqrt{3+3}}=9.4998$$ 1h

$$IS(C)=E_{1,3}\left(\frac{1\times 3}{1+3}\right)+E_{2,2}\left(\frac{2\times 2}{2+2}\right)+E_{2,3}\left(\frac{2\times 3}{2+3}\right)+E_{3,3}\left(\frac{3\times 3}{3+3}\right)=21.7$$ 1i

$$SD(C)=E_{1,3}(\frac{1}{3}+\frac{3}{1})+E_{2,2}(\frac{2}{2}+\frac{2}{2})+E_{2,3}(\frac{2}{3}+\frac{3}{2})+E_{3,3}(\frac{3}{3}+\frac{3}{3})=43.6667$$ 1j

$$F(C)=E_{1,3}(1^2+3^2)+E_{2,2}(2^2+2^2)+E_{2,3}(2^2+3^2)+E_{3,3}(3^2+3^2)=214$$ 1k

$$ABS(C)=E_{1,3}\sqrt{\frac{1+3-2}{1+3}}+E_{2,2}\sqrt{\frac{2+2-2}{2+2}}+E_{2,3}\sqrt{\frac{2+3-2}{2+3}}+E_{3,3}\sqrt{\frac{3+3-2}{3+3}}=14.7659$$ 1l

The remaining indices can be computed similarly as in eqs –, by applying the edge partition method to their respective definitions. A consolidated summary of the computed values for all topological indices is presented in Table .

4. Computed DTI Values for the Candidate H1-Antihistamines.

name	M 1	M 2	H	HM	ABC	R	GA	SCI	IS	SD	F	ABS
diphenhydramine	90	100	9.0667	414	14.2800	9.2709	19.6108	9.4998	21.7000	43.6667	214	14.7659
promethazine	106	126	9.3667	524	15.6823	9.6647	21.4364	10.1190	25.3500	49.3333	272	16.6432
hydroxyzine	128	146	12.5667	600	19.7871	12.7752	27.5866	13.2214	31.1167	59.6667	308	20.7398
pheniramine	86	96	8.5667	398	13.5729	8.7709	18.6108	8.9998	20.7000	41.6667	206	14.0587
cyproheptadine	120	144	10.6667	590	17.5849	10.8265	24.6640	11.5134	29.2500	53.0000	302	18.8995
meclizine	146	169	13.4000	700	22.0178	13.6867	30.4088	14.3802	35.2000	67.3333	362	23.3283
cinnarizine	142	163	13.8000	664	21.7990	13.8990	30.7576	14.5913	34.9000	64.0000	338	23.0584
chlorpheniramine	92	103	8.8667	432	14.3894	9.1647	19.4364	9.3942	21.8500	45.3333	226	14.9008
triprolidine	106	121	10.1667	498	16.2920	10.3265	22.6640	10.7886	25.7500	49.0000	256	17.1571
clemastine	124	145	11.2381	612	18.5988	11.5932	25.2725	12.0446	29.3619	59.0833	322	19.5545
fexofenadine	194	230	17.0619	978	28.7407	17.7242	38.6639	18.4007	45.5214	93.5000	518	30.2312
loratadine	144	172	12.8667	710	21.1893	13.1690	29.4122	13.8406	34.7667	65.3333	366	22.6172
desloratadine	120	144	10.6667	590	17.5849	10.8265	24.6640	11.5134	29.2500	53.0000	302	18.8995
cetirizine	134	152	12.8000	632	20.7129	13.1310	28.3556	13.5913	32.1500	64.0000	328	21.6441
azelastine	144	170	12.7667	706	21.3392	13.1142	29.3152	13.8022	34.5500	66.3333	366	22.7050
ebastine	180	206	16.3857	872	27.2733	16.8980	36.9325	17.5998	42.5643	86.8333	460	28.6186
rupatadine	164	195	14.3667	806	24.0582	14.6698	33.3684	15.5911	39.6000	73.6667	416	25.8034

2.2. Neighborhood Degree-Based Topological Indices

The indices defined in Table are derived from the neighborhood degree D _z of a vertex z, which is defined as the sum of the degrees of all vertices adjacent to z, i.e., $D_z=\sum _{z_1\in N(z)}{d}_{z_1}$ .

5. List of NDTIs Used in the Study.

additive NDTIs
neighborhood first Zagreb index	$$N{M}_1(G)=\sum _{z_1\in V(G)}{[D_{z_1}]}^2$$
neighborhood forgotten index	$$NF(G)=\sum _{z_1\in V(G)}{[D_{z_1}]}^3$$
neighborhood second Zagreb index	$$N{M}_2(G)=\sum _{z_1{z}_2\in E(G)}[D_{z_1}\times D_{z_2}]$$
neighborhood harmonic index	$$NH(G)=\sum _{z_1{z}_2\in E(G)}\frac{2}{D_{z_1}+D_{z_2}}$$
neighborhood inverse-sum index	$$NI(G)=\sum _{z_1{z}_2\in E(G)}\frac{D_{z_1}\times D_{z_2}}{D_{z_1}+D_{z_2}}$$

multiplicative NDTIs
fifth multiplicative first Zagreb index	$$\mathrm{\Pi }N{M}_1(G)=\prod _{z_1{z}_2\in E(G)}[D_{z_1}+D_{z_2}]$$
fifth multiplicative second Zagreb index	$$\mathrm{\Pi }N{M}_2(G)=\prod _{z_1{z}_2\in E(G)}[D_{z_1}\times D_{z_2}]$$
multiplicative total neighborhood index	$$\mathrm{\Pi }TN(G)=\prod _{z_1\in V(G)}{D}_{z_1}$$
multiplicative first neighborhood index	$$\mathrm{\Pi }NM(G)=\prod _{z_1\in V(G)}{[D_{z_1}]}^2$$
multiplicative F 1 neighborhood index	$$\mathrm{\Pi }NF(G)=\prod _{z_1\in V(G)}{[D_{z_1}]}^3$$

The vertex and edge partitions based on neighborhood degree sums, which are used in the computation of NDTIs, are presented in Tables and , respectively.

6. Vertex Partition of the Candidate Drugs Based on Neighborhood Degree Sum.

	D u
	2	3	4	5	6	7	8	9	10
diphenhydramine	0	2	8	6	0	2	1	0	0
promethazine	0	3	4	5	3	2	3	0	0
hydroxyzine	1	2	6	12	1	3	0	1	0
pheniramine	0	2	7	6	0	2	1	0	0
cyproheptadine	0	1	4	11	0	3	2	1	0
meclizine	0	2	4	14	4	3	0	1	0
cinnarizine	0	0	10	12	2	3	0	1	0
chlorpheniramine	0	3	4	9	0	2	1	0	0
triprolidine	0	1	5	11	1	2	1	0	0
clemastine	0	2	6	10	2	1	2	1	0
fexofenadine	0	3	10	14	3	1	5	0	1
loratadine	1	3	2	12	2	4	2	1	0
desloaratadine	0	1	5	9	1	3	2	1	0
cetirizine	0	3	6	13	1	3	0	1	0
azelastine	0	3	3	12	3	3	3	0	0
ebastine	0	1	10	14	5	3	2	0	0
rupatadine	0	2	3	14	5	3	2	1	0

7. Edge Partition of the Candidate Drugs Based on Neighborhood Degree Sum.

Diphenhydramine
D u , D v	3, 4	4, 4	4, 5	5, 7	5, 8	7, 8
no. of edges	2	4	7	4	1	2
Promethazine
D u , D v	3, 5	3, 6	4, 4	4, 5	5, 6	5, 7	5, 8	6, 6	6, 7	6, 8	7, 8	8, 8
no. of edges	2	1	2	4	1	2	2	1	2	1	2	2
Hydroxyzine
D u , D v	2, 3	3, 4	3, 5	4, 4	4, 5	5, 5	5, 6	5, 7	5, 9	7, 9
no. of edges	1	1	1	4	3	6	3	5	1	3
Pheniramine
D u , D v	3, 4	4, 4	4, 5	5, 5	5, 7	5, 8	7, 8
no. of edges	2	4	5	1	4	1	2
Cyproheptadine
D u , D v	3, 5	4, 4	4, 5	5, 5	5, 7	5, 8	7, 8	7, 9	8, 9
no. of edges	1	2	4	5	6	2	2	1	2
Meclizine
D u , D v	3, 5	4, 4	4, 5	5, 5	5, 6	5, 7	6, 6	7, 9
no. of edges	2	2	4	7	5	6	2	3
Cinnarizine
D u , D v	4, 4	4, 5	5, 5	5, 6	5, 7	7, 9
no. of edges	6	8	2	6	6	3
Chlorpheniramine
D u , D v	3, 4	3, 5	4, 4	4, 5	5, 5	5, 7	5, 8	7, 8
no. of edges	2	1	2	3	5	4	1	2
Triprolidine
D u , D v	3, 5	4, 4	4, 5	5, 5	5, 6	5, 7	5, 8	7, 8
no. of edges	1	3	4	5	3	4	1	2
Clemastine
D u , D v	3, 5	3, 6	4, 4	4, 5	4, 6	4, 9	5, 5	5, 6	5, 7	5, 8	6, 7	6, 9	8, 9
no. of edges	1	1	2	5	1	1	4	1	2	4	1	1	2
Fexofenadine
D u , D v	3, 6	4, 4	4, 5	4, 6	4, 8	4, 10	5, 5	5, 6	5, 7	5, 8	6, 7	8, 8	8, 10
no. of edges	3	4	6	1	2	1	4	4	2	8	1	1	3
Loratadine
D u , D v	2, 3	3, 5	3, 6	4, 4	4, 5	5, 5	5, 6	5, 7	5, 8	6, 7	7, 8	7, 9	8, 9
no. of edges	1	2	1	1	2	5	2	7	2	2	2	1	2
Desloaratadine
D u , D v	3, 5	4, 4	4, 5	5, 5	5, 6	5, 7	5, 8	6, 7	7, 8	7, 9	8, 9
no. of edges	1	3	4	3	1	5	2	1	2	1	2
Cetirizine
D u , D v	3, 4	3, 5	4, 4	4, 5	5, 5	5, 6	5, 7	7, 9
no. of edges	2	1	3	5	6	3	6	3
Azelastine
D u , D v	3, 5	3, 7	4, 4	4, 5	5, 5	5, 6	5, 7	5, 8	6, 6	6, 7	6, 8	7, 8	8, 8
no. of edges	2	1	1	4	7	2	2	2	1	2	1	4	1
Ebastine
D u , D v	3, 6	4, 4	4, 5	4, 6	5, 5	5, 6	5, 7	5, 8	6, 6	6, 7	6, 8	7, 8
no. of edges	1	4	6	3	4	6	6	2	1	1	2	2
Rupatadine
D u , D v	3, 5	4, 4	4, 5	5, 5	5, 6	5, 7	5, 8	6, 6	6, 7	7, 8	7, 9	8, 9
no. of edges	2	1	4	6	5	5	2	3	1	2	1	2

For illustration, Theorem 2 provides a detailed computation of the NDTIs for diphenhydramine.

Theorem 2. For the chemical graph C of diphenhydramine, NM ₁(C) = 458, NM ₂(C) = 520, NF(C) = 2514, NI(C) = 49.194, NH(C) = 4.214, ΠNM ₁(C) = 5.82 × 10¹⁹, ΠNM ₂(C) = 2.27 × 10²⁷, ΠTN(C) = 3.61 × 10¹², ΠNM(C) = 1.31 × 10²⁵, and ΠNF(C) = 4.72 × 10³⁷.

Proof. Using the definitions from Table and the vertex and edge partitions from Tables and

$$N{M}_1(C)=2\times 3^2+8\times 4^2+6\times 5^2+2\times 7^2+1\times 8^2=458$$ 2a

$$N{M}_2(C)=2[3\times 4]+4[4\times 4]+7[4\times 5]+4[5\times 7]+1[5\times 8]+2[7\times 8]=520$$ 2b

$$NF(C)=2\times 3^3+8\times 4^3+6\times 5^3+2\times 7^3+1\times 8^3=2514$$ 2c

$$NI(C)=2\left(\frac{3\times 4}{3+4}\right)+4\left(\frac{4\times 4}{4+4}\right)+7\left(\frac{4\times 5}{4+5}\right)+4\left(\frac{5\times 7}{5+7}\right)+1\left(\frac{5\times 8}{5+8}\right)+2\left(\frac{7\times 8}{7+8}\right)=49.194$$ 2d

$$NH(C)=2\left(\frac{2}{3+4}\right)+4\left(\frac{2}{4+4}\right)+7\left(\frac{2}{4+5}\right)+4\left(\frac{2}{5+7}\right)+1\left(\frac{2}{5+8}\right)+2\left(\frac{2}{7+8}\right)=4.214$$ 2e

$$\mathrm{\Pi }N{M}_1\left(C\right)={[3+4]}^2\times {[4+4]}^4\times {[4+5]}^7\times {[5+7]}^4\times {[5+8]}^1\times {[7+8]}^2=5.82\times 10^{19}$$ 2f

$$\mathrm{\Pi }N{M}_2\left(C\right)={[3\times 4]}^2\times {[4\times 4]}^4\times {[4\times 5]}^7\times {[5\times 7]}^4\times {[5\times 8]}^1\times {[7\times 8]}^2=2.27\times 10^{27}$$ 2g

$$\mathrm{\Pi }TN(C)=3^2\times 4^8\times 5^6\times 7^2\times 8^1=3.61\times 10^{12}$$ 2h

$$\mathrm{\Pi }NM(C)={(3^2)}^2\times {(4^2)}^8\times {(5^2)}^6\times {(7^2)}^2\times {(8^2)}^1=1.31\times 10^{25}$$ 2i

$$\mathrm{\Pi }NF(C)={(3^3)}^2\times {(4^3)}^8\times {(5^3)}^6\times {(7^3)}^2\times {(8^3)}^1=4.72\times 10^{37}$$ 2j

The indices for the other drugs can be determined in a similar manner as in eqs – by applying the vertex and edge partition techniques in accordance with their definitions. A comprehensive summary of the computed NDTI values for all drugs is given in Table .

8. Computed NDTI Values for the Candidate H1-Antihistamines.

name	NM 1	NF	NM 2	NI	NH	ΠNM 1	ΠNM 2	ΠTN	ΠNM	ΠNF
diphenhydramine	458	2514	520	49.194	4.214	5.82 × 1019	2.27 × 1027	3.61 × 1012	1.31 × 1025	4.72 × 1037
promethazine	614	3832	748	61.710	4.068	1.06 × 1023	3.82 × 1032	1.17 × 1014	1.37 × 1028	1.60 × 1042
hydroxyzine	682	3920	806	72.248	5.699	1.59 × 1028	2.21 × 1039	3.33 × 1017	1.11 × 1035	3.71 × 1052
pheniramine	442	2450	505	47.250	3.970	7.19 × 1018	1.42 × 1026	9.03 × 1011	8.16 × 1023	7.37 × 1035
cyproheptadine	704	4440	861	70.792	4.574	1.76 × 1026	1.81 × 1037	7.41 × 1015	5.49 × 1031	4.07 × 1047
meclizine	804	4682	938	83.088	5.906	7.62 × 1031	8.14 × 1044	5.63 × 1019	3.17 × 1039	1.78 × 1059
cinnarizine	760	4330	885	80.454	6.144	2.45 × 1031	8.99 × 1043	2.84 × 1019	8.09 × 1038	2.30 × 1058
chlorpheniramine	478	2660	548	50.680	4.075	1.11 × 1020	8.13 × 1027	5.29 × 1012	2.80 × 1025	1.48 × 1038
triprolidine	562	3136	650	59.656	4.522	2.17 × 1023	4.88 × 1032	3.53 × 1014	1.24 × 1029	4.39 × 1043
clemastine	694	4216	825	70.325	4.890	9.02 × 1026	8.26 × 1037	5.23 × 1016	2.73 × 1033	1.43 × 1050
fexofenadine	1114	7022	1322	109.846	7.380	6.67 × 1041	8.41 × 1058	8.56 × 1025	7.33 × 1051	6.28 × 1077
loratadine	840	5274	1023	84.256	5.589	2.40 × 1031	2.54 × 1044	1.05 × 1019	1.10 × 1038	1.16 × 1057
desloaratadine	706	4470	864	70.834	4.593	1.68 × 1026	1.67 × 1037	7.11 × 1015	5.06 × 1031	3.60 × 1047
cetirizine	712	4064	826	74.909	5.803	1.93 × 1029	8.58 × 1040	2.50 × 1018	6.25 × 1036	1.56 × 1055
azelastine	822	4986	988	83.504	5.519	2.27 × 1031	2.50 × 1044	1.60 × 1019	2.56 × 1038	4.10 × 1057
ebastine	974	5550	1130	101.105	7.227	1.20 × 1039	9.55 × 1054	3.28 × 1024	1.07 × 1049	3.52 × 1073
rupatadine	952	5858	1150	96.118	6.170	5.32 × 1035	5.76 × 1050	5.40 × 1021	2.92 × 1043	1.58 × 1065

3. QSPR Analysis of H1-Antihistamines

A quantitative structure–property relationship (QSPR) analysis was conducted on the selected set of H1-antihistamines to investigate the correlation between the molecular structure and key physicochemical properties. The properties considered include boiling point at 760 mmHg (BP in °C), enthalpy of vaporization (EV in kJ/mol), flash point (FP in °C), molar refraction (MR in cm³/mol), polarizability (α in ų), and molar volume (MV in cm³/mol). The corresponding values were sourced from ChemSpider and are summarized in Table .

9. Physicochemical Properties of the H1-Antihistamines.

sl. no.	name	BP	EV	FP	MR	α	MV
1	diphenhydramine	343.7	58.8	101.5	79.6	31.5	249.2
2	promethazine	403.7	65.5	198	87.8	34.8	251.3
3	hydroxyzine	499.2	80.8	255.7	105.9	42	317.1
4	pheniramine	348.3	59.3	164.5	75.9	30.1	236.1
5	cyproheptadine	440.1	69.7	194.5	91.6	36.3	257.5
6	meclizine	495.3	76.3	253.3	118	46.8	337.2
7	cinnarizine	509.2	78	229.8	119.3	47.3	337.2
8	chlorpheniramine	379	62.7	183	80.8	32	248
9	triprolidine	435.4	69.2	217.1	88.1	34.9	262.2
10	clemastine	425.2	68	211	100.4	39.8	313.3
11	fexofenadine	697.3	107.2	375.5	145.9	57.8	428.1
12	loratadine	531.3	80.7	275.1	105.9	42	303.5
13	desloratadine	467.9	73	236.8	90.1	35.7	254.4
14	cetirizine	542.1	86.3	281.6	105.9	42	314.2
15	azelastine	533.9	81	276.7	110	43.6	304.6
16	ebastine	596.3	88.8	314.5	144.7	57.3	428.6
17	rupatadine	586.4	87.6	308.4	122.4	48.5	337.2

To evaluate the predictive accuracy and robustness of the regression models, a set of standard statistical metrics is utilized. These include the coefficient of determination (r ²), root-mean-square error (RMSE), mean absolute error (MAE), standard error (SE), F-statistic, and p-value. In addition to these quantitative measures, graphical validation is carried out through scatter plots to assess the fit between predicted and observed values and residual plots to examine the distribution and patterns of prediction errors.

3.1. Linear Regression Using DTIs

An initial assessment of the strength of association between each physicochemical property and the set of DTIs was conducted by using the correlation coefficient (r). The resulting correlation values are summarized in Table . For each property, the topological index exhibiting the highest correlation is highlighted, thereby indicating promising candidates for subsequent predictive modeling.

10. Correlation Coefficient Values Connecting Physicochemical Properties with DTIs.

	BP	EV	FP	MR	α	MV
M 1	0.9671	0.9401	0.9250	0.9776	0.9777	0.9370
M 2	0.9644	0.9341	0.9256	0.9578	0.9579	0.9070
H	0.9567	0.9381	0.9015	0.9899	0.9901	0.9646
HM	0.9625	0.9350	0.9263	0.9608	0.9608	0.9175
ABC	0.9639	0.9420	0.9170	0.9897	0.9898	0.9603
R	0.9583	0.9416	0.9061	0.9914	0.9915	0.9700
GA	0.9640	0.9384	0.9132	0.9859	0.9860	0.9479
SCI	0.9617	0.9402	0.9098	0.9899	0.9900	0.9598
IS	0.9670	0.9369	0.9214	0.9743	0.9744	0.9264
SD	0.9563	0.9400	0.9174	0.9835	0.9835	0.9667
F	0.9588	0.9338	0.9249	0.9614	0.9613	0.9247
ABS	0.9661	0.9402	0.9200	0.9850	0.9851	0.9475

To evaluate the predictive capability of the TIs, linear regression analysis was conducted for each physicochemical property using the index that exhibited the highest correlation. The employed regression model follows the form y = z ₀ + z ₁ X, where y represents the physicochemical property, z ₀ is the intercept, and z ₁ is the regression coefficient corresponding to topological index X. A comprehensive statistical summary of the regression parameters for the most strongly correlated DTIs is presented in Table .

11. Regression Analysis of the Physicochemical Properties Related to DTIs.

	BP–M 1	EV–ABC	FP–HM	MR–R	α–R	MV–R
r 2	0.9353	0.8873	0.8580	0.9829	0.9831	0.9408
adjusted r 2	0.9310	0.8798	0.8485	0.9817	0.9820	0.9369
RMSE	23.1821	4.0491	23.6185	2.6598	1.0471	13.7707
MAE	19.0942	3.0775	18.9219	2.1025	0.8327	11.3949
SE	24.6792	4.3106	25.1438	2.8316	1.1148	14.6601
F-statistic	216.9	118.1	90.61	861.2	872.5	238.5
p-value	2.51 × 10–10	1.66 × 10–8	9.52 × 10–8	1.15 × 10–14	1.05 × 10–14	1.29 × 10–10
z 0	95.8348	23.3720	1.5424	6.6682	2.6475	38.7986
z 1	2.9753	2.6741	0.3777	7.9182	3.1377	21.5746

Accordingly, the best-fitting regression equations for each physicochemical property are summarized in eqs –, with the corresponding regression curves illustrated in Figure , and the associated residual plots are shown in Figure , collectively demonstrating the quality of fit achieved by the selected topological descriptors.

$$(\mathrm{B}\mathrm{P})=2.9753(M_1)+95.835$$ 3a

$$(\mathrm{E}\mathrm{V})=2.6741(ABC)+23.372$$ 3b

$$(\mathrm{F}\mathrm{P})=0.3777(HM)+1.5424$$ 3c

$$(\mathrm{M}\mathrm{R})=7.9182(R)+6.6682$$ 3d

$$(\alpha )=3.1377(R)+2.6475$$ 3e

$$(\mathrm{M}\mathrm{V})=21.575(R)+38.799$$ 3f

2.

Best-fit regression curves for the physicochemical properties with respect to DTIs.

3.

Residual plots for best-fit models with respect to DTIs.

3.2. Linear Regression Using Additive NDTIs

Similar to the approach used for DTIs, a preliminary analysis was conducted using correlation coefficients to examine the relationship between physicochemical properties and additive NDTIs. The resulting values are presented in Table .

12. Correlation Coefficient Values Connecting Physicochemical Properties with additive NDTIs.

	BP	EV	FP	MR	α	MV
NM 1	0.9561	0.9233	0.9215	0.9358	0.9360	0.8775
NF	0.9271	0.8950	0.8983	0.8751	0.8753	0.8067
NM 2	0.9431	0.9077	0.9125	0.9059	0.9061	0.8383
NI	0.9638	0.9306	0.9249	0.9584	0.9585	0.9039
NH	0.9375	0.9264	0.8778	0.9833	0.9834	0.9708

Based on the highest correlations, linear regression analysis was performed (see Table ), followed by the formulation of the corresponding regression equations (see eqs –).

13. Regression Analysis of the Physicochemical Properties Related to Additive NDTIs.

	BP–NI	EV–NI	FP–NI	MR–NH	α–NH	MV–NH
r 2	0.9288	0.8661	0.8554	0.9668	0.9672	0.9425
adjusted r 2	0.9241	0.8571	0.8457	0.9646	0.9650	0.9387
RMSE	24.3200	4.4138	23.8319	3.7020	1.4596	13.5694
MAE	19.3256	3.5384	19.4738	2.9461	1.1612	11.2860
SE	25.8906	4.6989	25.3710	3.9411	1.5539	14.4458
F-statistic	195.7	97	88.73	437.3	441.8	246.1
p-value	5.17 × 10–10	6.10 × 10–8	1.09 × 10–7	1.65 × 10–12	1.53 × 10–12	1.03 × 10–10
z 0	106.4276	27.7669	–9.5230	1.5493	0.6167	22.2812
z 1	5.0752	0.6484	3.3483	19.3262	7.6589	53.1417

The regression curves corresponding to the best-fitting regression equations are shown in Figure , and the residual plots are shown in Figure , illustrating the quality of fit achieved by the selected topological descriptors.

$$(\mathrm{B}\mathrm{P})=5.0752(NI)+106.4276$$ 4a

$$(\mathrm{E}\mathrm{V})=0.6484(NI)+27.7669$$ 4b

$$(\mathrm{F}\mathrm{P})=3.3483(NI)-9.5230$$ 4c

$$(\mathrm{M}\mathrm{R})=19.3262(NH)+1.5493$$ 4d

$$(\alpha )=7.6589(NH)+0.6167$$ 4e

$$(\mathrm{M}\mathrm{V})=53.1417(NH)+22.2812$$ 4f

4.

Best-fit regression curves for the physicochemical properties with respect to additive NDTIs.

5.

Residual plots for best-fit models with respect to additive NDTIs.

3.3. Logarithmic Regression Using Multiplicative NDTIs

In the case of multiplicative NDTIs, it is observed that their natural logarithms exhibit stronger agreement with the physicochemical properties compared to the original values. Additionally, the raw index values are considerably large, making their logarithmic forms more manageable for analysis and visualization. The corresponding natural logarithmic values are listed in Table , and their correlations with the physicochemical properties are presented in Table .

14. Natural Logarithm Values for the Multiplicative NDTIs.

name	ln(ΠNM 1)	ln(ΠNM 2)	ln(ΠTN)	ln(ΠNM)	ln(ΠNF)
diphenhydramine	45.51	62.99	28.92	57.83	86.75
promethazine	53.02	75.02	32.39	64.79	97.18
hydroxyzine	64.93	90.59	40.35	80.70	121.04
pheniramine	43.42	60.22	27.53	55.06	82.59
cyproheptadine	60.43	85.79	36.54	73.08	109.62
meclizine	73.41	103.41	45.48	90.95	136.43
cinnarizine	72.27	101.21	44.79	89.59	134.38
chlorpheniramine	46.16	64.27	29.30	58.59	87.89
triprolidine	53.73	75.27	33.50	66.99	100.49
clemastine	62.07	87.31	38.49	76.99	115.48
fexofenadine	96.30	135.68	59.71	119.42	179.14
loratadine	72.26	102.25	43.80	87.60	131.39
desloaratadine	60.39	85.71	36.50	73.00	109.50
cetirizine	67.43	94.25	42.36	84.73	127.09
azelastine	72.20	102.23	44.22	88.44	132.66
ebastine	89.99	126.60	56.45	112.90	169.35
rupatadine	82.26	116.88	50.04	100.08	150.12

15. Correlation Coefficient Values Connecting Physicochemical Properties with Natural Logarithm of the Multiplicative NDTIs.

	BP	EV	FP	MR	α	MV
ln(ΠNM 1)	0.9672	0.9394	0.9232	0.9789	0.9790	0.9369
ln(ΠNM 2)	0.9664	0.9365	0.9239	0.9742	0.9743	0.9284
ln(ΠTN)	0.9643	0.9405	0.9188	0.9879	0.9879	0.9557
ln(ΠNM)	0.9643	0.9405	0.9188	0.9879	0.9879	0.9557
ln(ΠNF)	0.9643	0.9405	0.9188	0.9879	0.9879	0.9557

Based on the most strongly correlated pairs, logarithmic regression analysis was performed using the model y = z ₀ + z ₁ ln­(X), where y represents the physicochemical property, z ₀ is the intercept, and z ₁ is the regression coefficient associated with the natural logarithm of the multiplicative NDTI­(X). The results of this analysis are summarized in Table , and the corresponding regression equations are presented in eqs –.

$$(\mathrm{B}\mathrm{P})=6.0161\ln (\mathrm{\Pi }N{M}_1)+89.5110$$ 5a

$$(\mathrm{E}\mathrm{V})=1.2664\ln (\mathrm{\Pi }TN)+24.6252$$ 5b

$$(\mathrm{F}\mathrm{P})=2.7673\ln (\mathrm{\Pi }N{M}_2)-15.6917$$ 5c

$$(\mathrm{M}\mathrm{R})=2.2420\ln (\mathrm{\Pi }TN)+13.2073$$ 5d

$$(\alpha )=0.8884\ln (\mathrm{\Pi }TN)+5.2408$$ 5e

$$(\mathrm{M}\mathrm{V})=6.0403\ln (\mathrm{\Pi }TN)+59.3899$$ 5f

16. Regression Analysis of the Physicochemical Properties Related to Multiplicative NDTIs.

	BP–ln(ΠNM 1)	EV–ln(ΠTN)	FP–ln(ΠNM 2)	MR–ln(ΠTN)	α–ln(ΠTN)	MV–ln(ΠTN)
r 2	0.9354	0.8845	0.8537	0.9759	0.9760	0.9134
adjusted r 2	0.9311	0.8768	0.8439	0.9743	0.9744	0.9076
RMSE	23.1651	4.0987	23.9735	3.1572	1.2482	16.6644
MAE	19.1819	3.0967	20.1366	2.7030	1.0705	14.0778
SE	24.6611	4.3634	25.5218	3.3611	1.3288	17.7406
F-statistic	217.3	114.9	87.5	606.8	609.6	158.1
p-value	2.48 × 10–10	1.99 × 10–8	1.19 × 10–7	1.51 × 10–13	1.46 × 10–13	2.28 × 10–9
z 0	89.5110	24.6252	–15.6917	13.2073	5.2408	59.3899
z 1	6.0161	1.2664	2.7673	2.2420	0.8884	6.0403

The scatter plots corresponding to the best-fitting models, based on the natural logarithm of the respective multiplicative NDTIs, along with the associated residual plots, are presented in Figures and , respectively.

6.

Best-fit regression curves for the physicochemical properties with respect to multiplicative NDTIs.

7.

Residual plots for best-fit models with respect to multiplicative NDTIs.

3.4. Prediction of Physicochemical Properties

To assess the generalizability of the developed models, two additional H1-antihistaminescyclizine and doxylaminewere analyzed. Their molecular structures are shown in Figure , and the corresponding TIs, computed, as described in Theorems 1 and 2, are listed in Table . Using the previously established regression models (eqs –, –, and –), predictions for their physicochemical properties were generated and compared with the actual values. The results are illustrated in Figure , along with a comparison table where the predicted values from model 1 (eqs –), model 2 (eqs –), and model 3 (eqs –) are displayed.

8.

Chemical structures of additional antihistamines.

17. Selected TIs for Cyclizine and Doxylamine.

	M 1	ABC	HM	R	NI	NH	ln(ΠNM 1)	ln(ΠNM 2)	ln(ΠTN)
cyclizine	102	15.5444	484	9.8433	58.076	4.314	51.62	72.42	31.94
doxylamine	98	15.1037	476	9.6268	54.309	4.243	48.89	68.09	30.87

9.

Comparison between predicted and actual values of the physicochemical properties for cyclizine and doxylamine with respect to the derived regression models, namely model 1 (eqs –), model 2 (eqs –), and model 3 (eqs –).

4. Discussion

All of the derived models demonstrated high r ² values, almost every value exceeding 0.90, indicating that the selected TIs explain a substantial proportion of the variability in the physicochemical properties. The adjusted r ², which corrects for the number of predictors, remained closely aligned with the unadjusted r ², confirming that the models are not overfitted and that the descriptors are relevant contributors.

Error-based metrics, such as RMSE and MAE, were consistently low, highlighting the minimal average deviation between predicted and observed values. RMSE reflects the standard deviation of prediction errors, while MAE captures the average absolute difference, both confirming the high predictive accuracy of the models. Similarly, the SE values remained low across the models, suggesting a strong fit of the data to the regression line.

The F-statistic values, which assess the overall significance of the regression, were notably high with corresponding p-values far below the conventional significance level of 0.05. This confirms that the models are statistically significant and that the TIs play a meaningful role in predicting the molecular properties.

In addition to numerical measures, graphical tools were employed to visually validate the models. Scatter plots confirmed the linearity and strength of the associations, with data points closely clustered around the ideal fit line. Residual plots were also examined to assess the distribution of the prediction errors. The absence of discernible patterns or systematic deviations in these plots suggests that model assumptions, such as homoscedasticity and linearity, are reasonably satisfied.

Among the physicochemical properties examined, the models defined in eqs –, –, and – provide the best predictions for polarizability (α), MR, MV, and BP, as indicated by their high r ² values, averaging 0.975, 0.975, 0.932, and 0.933, respectively. EV and FP exhibit moderate but acceptable correlations, with mean r ² values of 0.879 and 0.856.

Among all TIs considered, the Randi index consistently exhibits strong predictive power across multiple DTIs, particularly for MR, α, and MV, where it achieves near-unity correlation coefficients (refer Table ). This highlights the Randić index as a reliable and effective structural descriptor. In the class of additive NDTIs, the NI and NH indices demonstrate superior predictive capabilities (refer Table ). These findings underline the fact that reverse degree indices perform exceptionally well among the evaluated descriptors, as they not only exhibit stronger statistical correlations with physicochemical properties but also capture subtle structural variations more effectively than conventional indices. For the multiplicative NDTIs, all the considered indices, in particular ΠTN, ΠNM, and ΠNF, emerge as effective predictors for specific properties (refer Table ). The multiplicative NDTIs outperformed the additive NDTIs in all cases except for the prediction of the MV. Although DTIs generally showed superior performance in the analysis, their advantage over the multiplicative NDTIs was marginal, indicating that the two sets of descriptors exhibit nearly equivalent predictive capability. The ability of NDTIs to highlight secondary structural effects makes them valuable for enriching QSPR models, as they contribute new insights into the relationship between molecular topology and chemical behavior, even when their standalone predictive strength appears weaker.

To assess the external validity of the derived models, two additional H1-antihistamines, namely, cyclizine and doxylamine, were analyzed. As shown in Figure , the predicted values align closely with the experimentally observed data, especially for MR, α, and MV. Although minor deviations are noted for other properties, the overall predictions remain within an acceptable margin, indicating that the models possess good generalizability.

Beyond statistical observations, meaningful chemical inferences can also be drawn from the data set. For instance, cyclizine is a tricyclic compound with relatively less branching, while doxylamine is a bicyclic compound exhibiting greater branching (Figure ). This structural difference is reflected in the predictive performance of the models (Figure ): properties such as BP, EV, and FP are more accurately predicted for doxylamine, whereas MR shows better agreement with actual values in the case of cyclizine. Interestingly, both compounds exhibit close predictive-experimental correspondence for parameters like MR and polarizability. These findings suggest that the number of cycles in the molecular framework and the extent of branching play a significant role in determining how well certain physicochemical properties can be modeled, offering valuable insights into the interplay between structure and property prediction in QSPR analysis.

These results affirm the utility of TIs in modeling structure–property relationships. By leveraging topological descriptors within the QSPR paradigm, researchers can efficiently screen and optimize drug candidates in the early stages of development, significantly reducing the need for extensive experimental trials.

5. Conclusions

In this study, a QSPR-based approach utilizing DTIs and NDTIs was employed to model and predict key physicochemical properties of H1-antihistamines. Overall, the combination of strong statistical indicators and supportive graphical validation demonstrates that the developed models are both accurate and generalizable, making TIs effective descriptors in the QSPR analysis of the selected drug compounds. For NDTIs, although the observed correlation coefficients with physicochemical properties were relatively lower compared to conventional degree-based indices, they provide complementary values by offering additional structural sensitivity. Validation using external compounds, cyclizine and doxylamine, confirmed the generalizability of the model. Also, the analysis highlights that the number of cycles and the degree of branching in molecular structures critically influence the accuracy of the QSPR property predictions. Overall, this study demonstrates that computational analysis using topological descriptors not only reduces experimental effort but also offers superior insights into the molecular factors influencing drug behavior, thereby supporting the rational design of next-generation antihistamines.

This work can be extended by exploring other types of indices, such as domination-based TIs, and by modeling additional physicochemical or pharmacokinetic properties. Such analyses can be conducted not only on the same class of drugs but also across other classes of drugs or chemical compounds, thereby broadening the applicability and scope of the QSPR framework.

All chemical structures used in this study were retrieved from the PubChem database (https://pubchem.ncbi.nlm.nih.gov/). The corresponding physicochemical properties were obtained from the ChemSpider database (https://www.chemspider.com/). Statistical analyses and graphical representations were performed using Python and Microsoft Excel. All relevant data are provided within the manuscript.

The authors declare no competing financial interest.