Quantitative Structure–Property Relationship Analysis in Molecular Graphs of Some Anticancer Drugs with Temperature Indices Approach

: The most important application of anticancer drugs in various forms (alkylating agents, hormones agents, and antimetabolites) is the treatment of malignant diseases. Topological indices are widely used in the ﬁeld of chemical and medical sciences, especially in studying the chemical, biological, clinical, and therapeutic aspects of drugs. In this article, the temperature indices in anticancer drugs molecular graphs such as Carmustine, Convolutamine F, Raloxifene, Tambjamine K, and Pterocellin B were calculated and then analyzed based on physical and chemical properties. The analysis was performed by identifying the best regression models based on temperature indices for six physical and chemical features of anticancer drugs. The results indicated that temperature indices were essential topological indices that predict the properties of anticancer drugs, such as boiling point, ﬂash point, enthalpy, molar refractivity, molar volume, and polarizability. It was also observed that the r value of the regression model was more than 0.6, and the p value was less than 0.05.


Introduction
The rapid growth of abnormal cells in the human body causes cancer.The drugs used to treat cancer usually include alkylates and metabolites.Making and researching primary drugs for cancer treatment are of interest to doctors, but the drug discovery process, from identifying novel chemical entities to establishing regulatory approval is complex, costly, and time consuming.Traditional methods often suffer from inefficiencies in compound synthesis and biological screening, which prompts the scientific community to seek faster and more efficient ways to discover lead compounds.
Graph theory of chemistry is an interdisciplinary science that is used to study molecular structures and create correlations between different activities, properties, and events.
The molecular graph is related to the structural formula of a chemical compound, and the vertices and edges of the graph correspond to the atoms and chemical bonds of the compound, respectively.Chemical graph theory has provided creative and unique tools for the study of chemical structures, among which topological indices can be mentioned.
The topological index is used as the descriptor to study the structure and some properties of a molecular graph and is expressed as a real number [1][2][3].The scarcity and expensiveness of anticancer drugs and the production of more effective drugs lead to studies on drug structures, especially using topological indices.
Many studies were conducted on molecular graphs, especially nanotubes, and pharmaceutical structures, using topological indices [4][5][6][7][8][9][10].Lokesha et al. studied the temperature indices of certain Archimedian Lattices [11].Adnan et al. investigated the degree-based indices on some anticancer drugs and found a high correlation between the indices and the physical-chemical properties of the drugs [12].Hayat et al. used temperature indices to predict the physicochemical properties of polycyclic aromatic hydrocarbons and benzenoid hydrocarbons as well as investigate the π-electron energy of benzenoid hydrocarbons [13][14][15].
One of the most important methods of studying the relationship between the physicalchemical properties and topological indices of a material is the application of QSPR (quantitative structure-property relationship analysis) models.In the QSPR model, the curve regression method is used as a tool to analyze the relationship between physical and chemical properties and topological indices.Various QSAR studies using topological indices have been performed on multiple drug structures [16,17].
In this article, different temperature indices are investigated on some anticancer drugs to be used for researchers to understand the physical characteristics and chemical reactions related to them.

Preliminaries
In the molecular graph G, the degree of arbitrary vertex s is denoted by d s , and the connection of two vertices z and w is expressed by zw ∈ E(G).

Definition 1.
The temperature of a vertex s ∈ E(G) is defined as follows [18]: where n is the number of graph vertices.

Temperature Index Name Definition
Recently, studies have been conducted on temperature indices [21,22].

Results
In this section, after introducing the molecular graphs of Carmustine, Convolutamine F, Raloxifene, Tambjamine K, Pterocellin B and obtaining the temperature of the vertices, the temperature indices are calculated.
Figure 1 shows the molecular graph of Carmustine.The molecular graph of Carmustine is represented by C.There are five types in C as follows: The study of the edges in C is shown in Table 2.
Using Tables 1 and 2, we can calculate the following topological indices for C • , ( ) Theorem 1. Assume that C is the molecular graph of Carmustine.Then, the temperatur of C are computed as follows: The molecular graph of Carmustine is represented by C.There are five types of edges in C as follows: The study of the edges in C is shown in Table 2.

□
The molecular graph of Raloxifene (Figure 2) is represented by R.There are four types of edges in R as follows:  The study of the edges in R is shown in Table 3.The study of the edges in R is shown in Table 3. Theorem 2. Assume that R is the molecular graph of Raloxifene.Then, the temperature indices of R are computed as follows: Proof of Theorem 2. By using the relations in Table 3 and placing them in Table 1, the proof is complete.
The molecular graph of Tambjamine K (Figure 3) is represented by T. There are five types of edges in T as follows: 31 31 Theorem 2. Assume that R is the molecular graph of Raloxifene.Then, the temperature indices of R are computed as follows: ,    3 and placing them in Table 1, the proof is complete.□ The molecular graph of Tambjamine K (Figure 3) is represented by T. There are five types of edges in T as follows:  The study of the edges in T is shown in Table 4.The study of the edges in T is shown in Table 4. Theorem 3. Assume that T is the molecular graph of Tambjamine K.Then, the temperature indices of T are computed as follows: Proof of Theorem 3. By using the relations in Table 4 and placing them in Table 1, the proof is complete.

E rs E T T T E rs E T T T E rs E T T T E rs E T T T E rs E T T T
The molecular graph of Pterocellin B (Figure 4) is represented by P.There are five types of edges in P as follows:  4 and placing them in Table 1, the proof is complete.□ The molecular graph of Pterocellin B (Figure 4) is represented by P.There are five types of edges in P as follows:  The study of the edges in P is shown in Table 5. Theorem 4. Assume that P is the molecular graph of Pterocellin B. Then, the temperature indices of P are computed as follows: Proof of Theorem 4. By using the relations in Table 5 and placing them in Table 1, the proof is complete.

E rs E P T T E rs E P T T E rs E P T T E rs E P T T E rs E P T T
The molecular graph of Convolutamine F (Figure 5) is represented by F. There are five types of edges in F as follows: 3  13 , T s = 3 13 .
The study of the edges in F is shown in Table 6.Theorem 5. Assume that F is the molecular graph of Convolutamine F.Then, the temperature indices of F are computed as follows: Proof of Theorem 5.By using the relations in Table 6 and placing them in Table 1, the proof is complete.

Temperature Indices and QSPR
Here, we have four topological indices based on temperature indices such as PT (G), ST (G), SDT (G), and TSO (G) to model five representative physical properties (BP, EN, FP, MR, MV, and PO) with five anticancer drugs: Carmustine, Convolutamine F, Raloxifene, Tambjamine K, and Pterocellin B. The values of these properties are taken from ChemSpider (Table 7).Topological indices based on the above degree and experimental values for the physical and chemical properties of five anticancer drugs are shown in Table 8, respectively.From the data in Table 9, it is clear that all data values are normally distributed.Therefore, a regression model is used for data acceptance and analysis.
The results of Theorems 1-5 are shown in Table 8.No topological index displays a satisfactory correlation with TSO (G).The boiling point (BP) is the temperature at which the addition of heat leads to the conversion of a liquid into its vapor without increasing the temperature.The enthalpy (EN) of a system is equal to the internal energy of the system plus the product of pressure and volume.The flash point (FP) is the lowest temperature at which sufficient flammable vapor is present to support ignition when an ignition source is applied.Molar refractivity (MR) is a measure of the total polarizability of one mole of a substance.The molar volume (MV) of a substance is the ratio of the volume of a substance to the amount of the substance at a given temperature and pressure.Polarizability (PO) usually refers to the tendency of a material in an electric field to acquire a proportional electric dipole moment.The physical properties of drugs used for anticancer drugs are shown in Table 7.

Conclusions
Table 9 and the graphs in Figure 6 indicate the correlated values of the physicochemical properties of anticancer drugs with the defined temperature indices.It can be seen that the index ST (G) = 0.984 shows a higher positive and significant correlation with boiling point (BP), and also ST (G) = 0.984 shows a more positive correlation with flash point (FP) than the others.In the case of enthalpy, PT (G) shows the highest correlation value, i.e., 0.983.The SDT (G) index with molar refractivity (MR), molar volume (MV) and polarizability (PO) provides the highest positive correlation with values of 0.974, 0.974 and 0.975 for the physical and chemical properties, respectively.The boiling point (BP) is the temperature at which the addition of heat leads to the conversion of a liquid into its vapor without increasing the temperature.The enthalpy (EN) of a system is equal to the internal energy of the system plus the product of pressure and volume.The flash point (FP) is the lowest temperature at which sufficient flammable vapor is present to support ignition when an ignition source is applied.Molar refractivity (MR) is a measure of the total polarizability of one mole of a substance.The molar volume (MV) of a substance is the ratio of the volume of a substance to the amount of the substance at a given temperature and pressure.Polarizability (PO) usually refers to the tendency of a material in an electric field to acquire a proportional electric dipole moment.The physical properties of drugs used for anticancer drugs are shown in Table 7.

Conclusions
Table 9 and the graphs in Figure 6 indicate the correlated values of the physicochemical properties of anticancer drugs with the defined temperature indices.It can be seen that the index ST (G) =0.984 shows a higher positive and significant correlation with boiling point (BP), and also ST(G) = 0.984 shows a more positive correlation with flash point (FP) than the others.In the case of enthalpy, PT(G) shows the highest correlation value, i.e., 0.983.The SDT(G) index with molar refractivity (MR), molar volume (MV) and polarizability (PO) provides the highest positive correlation with values of 0.974, 0.974 and 0.975 for the physical and chemical properties, respectively.

2 .
By using the relations in Table

Table 1 .
Temperature-based topological indices of G.

Table 2 .
Edge partition of C.

Table 2 .
Edge partition of C.

Table 3 .
Edge partition of R.

Table 4 .
Edge partition of T.

Table 5 .
Edge partition of P.

Table 6 .
Edge partition of F.

Table 7 .
Physical properties of drugs used for anticancer drugs.

Table 8 .
Temperature indices of drugs used for anticancer drugs.

Table 9 .
Correlation coefficients of physical properties of drugs.