# Topological Indices of Drugs Used in Rheumatoid Arthritis Treatment and Its QSPR Modeling

Authors: Saima Parveen; Fozia Bashir Farooq; Nadeem Ul Hassan Awan; Rakotondrajao Fanja; Muhammad Farooq Choudhary
Journal: Journal of Mathematics
 (2022)

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## Abstract

A topological index is a molecular descriptor derived from the molecular structure of a chemical substance. These indices can be used to analyze mathematical values and predict various physical properties of drugs. This article discusses tofacitinib, leflunomide, upadacinib, baricitinib, filgotinib, methotrexate, and other drugs used to treat rheumatoid arthritis, and the goal of the QSPR study is to calculate the relationship between the properties under investigation (e.g., boiling point, polarity, and molar volume) and molecular descriptors. Topological indices (TIs) were imposed on said drugs to calculate the correlation with physicochemical properties in this course.


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## Body

## 1. Introduction

Topological indices (TIs) are numeric descriptors obtained through a molecular graph in order to fully examine the drugs and are widely used in the investigation and prediction of many drugs’ physicochemical properties. There are various types of polynomials and topological indices that are extensively calculated, represent the chemical structure, and play an important role in chemical graph theory. Among these families, degree-based topological indices are extremely important and play an important role in chemical graph theory. Among these techniques, the QSPR approach correlates a molecule’s biological activity with its physicochemical properties using a variety of descriptors, and it is used on drugs used in the treatment of rheumatoid arthritis. The QSPR approach has recently been used to develop models to predict drug properties. QSAR is a drug design source that specifies a relationship between molecules’ physicochemical properties and biological activities that affect drug response. The augmented Zagreb index in [1] is the best predictor of alkane heat of formation. The ABC index and Randic index are useful for calculating drug bioactivity. These properties are being researched because of their substantial impact on bioactivity and drug transit in the human body. In this paper, we compute TIs for drugs used in the treatment of rheumatoid arthritis. Similarly, anti-RA drugs are chemical compounds with carefully defined topological indices and deliberate QSPR analysis. Using linear regression is highly correlated with the characteristic of RA drugs based on mathematical observation. Rheumatoid arthritis (RA) is a chronic, autoimmune, and inflammatory joint disease. Estimates from North America and Northern Europe range from 20 to 50 cases per 100,000 people. The prevalence of RA in developing countries is unknown [2–5]. Scientists are constantly looking for the latest ways to treat people suffering from RA. One method is to create new drugs. Drug detection remains difficult to work because it is expensive, needs much time, and is much more difficult in some situations. This disease will cause functional impairment, premature mortality, joint erosions, and decreased quality of life over time. This necessitates prompt screening, diagnosis, and treatment which helps in order to control the disease. This work investigates ten drugs medicines tofacitinib, leflunomide, upadacinib, baricitinib, filgotinib, methotrexate, prednisolone, sulfasalazine, azathioprine, and hydroxychloroquine that are secure and reliable remedies which are the need of the community well-being. Figure 1 depicts the chemical structure of said drugs.Figure 1
Structure of drugs. (a) Azathioprine. (b) Hydroxycholoquine. (c) Sulfasalazine. (d) Filgothinib. (e) Leflunomide. (f) Prednisolone. (g) Methotrexate. (h) Baricitinib. (i) Tofacitinib. (j) Upadacitinib.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)


## 2. Material and Method

Elements denote vertices in drug structure, and the corresponding bonds of atoms are referred to as edges. GraphG (V, E) is simple, finite, and connected, whereas V and E in the chemical graph are referred to as vertex and edge set, respectively. The degree of a vertex in a graph G is denoted by du and is the number of vertices adjacent to it. Degree-based topological indices used are given as follows.Definition 1.
TheABC indices [6] is(1)ABCG=∑uv∈EGdu+dv−2dudv.Definition 2.
The Randic indexRAG [7] is(2)RAG=∑uv∈EG1dudv.Definition 3.
The sum connectivity index [8] is(3)SG=∑uv∈EG1du+dv.Definition 4.
The GA index [9] of a molecular graph G is(4)GAG=∑uv∈EG2dudvdu+dv.Definition 5.
The first and second Zagreb indices [10] are(5)M1G=∑uv∈EGdu+dv,M2G=∑uv∈EGdudv.Definition 6.
The harmonic index [11] is(6)HG=∑uv∈EG2du+dv.Definition 7.
The hyper Zagreb index [12] is(7)HMG=∑uv∈EGdu+dv2.Definition 8.
The forgotten index [13] is(8)FG=∑uv∈EGdu2+dv2.
Physical property values are obtained from Chemspider. The data in Table1 show that they are normally distributed. As a result, the linear regression model is best to examine and use in this analysis. We recommend that the reader read the following research articles [1, 14–32] for more information on TIs.
The molecular formula of methotrexate is C20H22N8O5. A wide range of cancers and rheumatoid arthritis are treated with methotrexate. Lymphocytic leukemia is also treated with this medicine. The molecular formula of baricitinib is C16H17N7O2S. Baricitinib is used in severe rheumatoid arthritis. The molecular formula of tofacitinib is C16H20N6O. The molecular formula of Upadacitinib is C17H19F3N6O. Upadacitinib is used to cure moderate to severe RA and slow down disease progression. The molecular formula of azathioprine is C9H7N7O2S. Azathioprine has a long duration of action. The molecular formula of hydroxychloroquine is C18H26ClN3O and is helpful in RA treatment. The molecular formula of sulfasatazne is C18H14N4O5S. Ulcerative colitis and rheumatoid arthritis are treated with the anti-inflammatory drug sulfasalazine. The molecular formula of filgotinib is C21H23N5O3S. Methotrexate is used alone or in combination with filgotinib to treat rheumatoid arthritis. More than 50% of patients suffering from RA are incapable to accomplish the availability of numerous cures including disease-modifying anti-rheumatic drugs (DMARS) like methotrexate. The molecular formula of leflunomide is C12H9F3N2O2. The use of this medical treatment retards the progression of structural damage and improves physical function.Table 1
The TI values of drugs.
Name of drugF (G)ABC (G)RA (G)S (G)M2 (G)GA (G)M1 (G)H (G)HM (G)Azatjoprine41020.5412.0012.4517426.3814411.15758Hydroxycholoqoine91637.9221.4121.5935345.0727818.881622Sulfasatazne62631.2918.9419.3427320.6022017.601172Filgotinib105042.4123.5224.3842652.2632221.081902Leflumomide45421.6512.6012.8218426.8015211.43822Predisolane117642.4623.5924.1246951.5333620.812114Methotrexate90642.1924.9025.2937352.8830022.711652Bracitinib82434.1219.2119.8433442.3225617.331492Tofacitinib82833.8018.9419.3532540.8825216.851478Upadacinib90236.6616.3320.9836144.7427618.141624


## 3. Results and Discussion

TIs are carried out on RA drugs in this section. The relationship between QSPR analysis and TIs shows that properties under discussion are highly correlated as regards of the disease’s physicochemical properties. Tofacitinib, leflunomide, upadacinib, baricitinib, filgotinib, methotrexate, neoral, prednisolone, sulfasalazine, azathioprine, and hydroxychloroquine are the ten medications used in the RA analysis. Figure1 depicts the drug structure. For drug research, we use regression analysis calculations.


### 3.1. Regression Model

In this article, the tenacity of QSPR modelling is tested using a drug computable structure analysis of nine topological indices. Table2 shows the four physical properties of ten RA medications: polarity, molar volume (MV), refractive index (R), and complexity. We run the regression analysis for the drugs, and the validated linear regression model is as follows:(9)P=A+bTI.Table 2
Correlation coefficients.
Topological indexCorrelation coefficient ofPolarityRefractive indexMolar volumeComplexityABC (G)0.8720.8720.8980.709RA (G)0.9090.9090.9140.729S (G)0.9330.9330.9080.735GA (G)0.9070.9070.8970.761M1 (G)0.8180.8180.8670.709M2 (G)0.7540.7540.830.707HM (G)0.7280.7270.820.677F (G)0.7220.7210.8210.649H (G)0.9310.930.9090.755P denotes physicochemical property. Letters A, TI, and b represent the topological index, constant, and regression coefficient, respectively. The software packages Statistix, Python, and MATLAB are helpful in determining the results. The physiochemical properties of nine TIs of RA drugs are investigated using a linear QSPR model. For the aforementioned calculation, equation (1) is appropriate.Theorem 1.
LetG1 be a graph of Sulfasatazne. The TIs of G1 are(10)ABCG1=31.29,RAG1=18.95,SG1=19.34,GAG1=40.60,M1G1=220,M2G1=273,FG1=626,HG1=17.60,HMG1=1172.Proof.
LetG1 be the graph of sulfasatazne, with edge set E and Em,n partition of G1 vertices with degrees m and n. E1,2=2,, ,E1,3=13,E1,4=2,E2,3=6,E3,3=18, and E3,4=2.(i)
Using Definition 1 and the partitionsEm,n, we obtain(11)ABCG1=21+2−21×2+131+3−21×3+21+4−21×4+62+3−22×3+183+3−23×3+23+4−23×4=31.29.(ii)
Using Definition 2 and the partitionsEm,n, we obtain(12)RAG1=211×2+1311×3+211×4+612×3+1813×3+213×4=18.95.(iii)
Using Definition 3 and the partitionsEm,n, we obtain(13)SG1=211+2+1311+3+211+4+612+3+1813+3+213+4=19.34.(iv)
Using Definition 4 and the partitionsEm,n, we obtain(14)GAG1=21×21+2+131×31+3+21×41+4+62×32+3+183×33+3+23×43+4=40.60.(v)
Using Definition 5 and the partitionsEm,n, we obtain(15)M1G1=21+2+131+3+21+4+62+3+183+3+23+4=220.(iii)
Using Definition 5 and the partitionsEm,n, we obtain(16)M2G1=21×2+131×3+21×4+62×3+183×3+23×4=273.(vi)
Using Definition 6 and the partitionsEm,n, we obtain(17)HG1=211+2+1311+3+211+4+612+3+1813+3+213+4=17.60.(vii)
Using Definition 7 and the partitionsEm,n, we obtain(18)HMG1=21+22+131+32+21+42+62+32+182+42+23+42=1172.(viii)
Using Definition 8 and the partitionsEm,n, we obtain(19)FG1=21+4+131+9+21+16+64+9+184+16+29+9+29+16=626.Theorem 2.
LetG2 be graph of Upadacinib. The TIs of G2 are as follows:(20)ABCG2=36.66,RAG2=20.33,SG2=20.98,GAG2=44.74,M1G2=276,M2G2=361,FG2=902,HG2=18.14,HMG2=1624.Proof.
LetG2 be graph of upadacinib having edge set E and Em,n partition of G2 vertices with degrees m and n. With E1,3=7,E1,4=16,E2,3=4,E3,3=12,E3,4=4, and E4,4=6.(i)
Using Definition1 and the partitions Em,n, we obtain(21)ABCG2=71+3−21×3+161+4−21×4+42+3−22×3+123+3−23×3+43+4−23×4+614×4=36.66.(ii)
Using Definition2 and the partitions Em,n, we obtain(22)RAG2=711×3+1611×4+412×3+1213×3+413×4+614×4=20.33.(iii)
Using Definition3 and the partitions Em,n, we obtain(23)SG2=711+3+1611+4+412+3+1213+3+413+4+614+4=20.98.(iv)
Using Definition4 and the partitions Em,n, we obtain(24)GAG2=71×31+3+161×41+4+42×32+3+123×33+3+43×43+4+64×44+4=44.74.(v)
Using Definition5 and the partitions Em,n, we obtain(25)M1G2=∑uv∈EG1su+sv=71+3+161+4+42+3+123+3+43+4+64+4=276.(vi)
Using Definition5 and the partitions Em,n, we obtain(26)M2G2=71×3+161×4+42×3+123×3+43×4+64×4=361.(vii)
Using Definition5 and the partitions Em,n, we obtain(27)HG2=711+3+1611+4+412+3+1213+3+413+4+614+4=18.14.(viii)
Using Definition6 and the partitions Em,n, we obtain(28)HMG2=31+22+31+32+61+42+42+32+52+42+33+32+13+42+44+42=1624.(ix)
Using Definition7 and the partitions Em,n, we obtain(29)FG2=31+4+41+9+61+16+44+9+54+16+39+9+19+16+416+16=902.
The remaining drugs topological indices can be calculated using the same procedure as applied in Theorem2, Theorem 1, and Definitions 1–8. Table 1 contains values for each drug.
Applying (1), we calculated the linear models for all TIs written as follows:(1)
Regression models forABC (G) are as follows:Polarity = 10.364 + 0.781 [ABC (G)]Refractive index = 26.174 + 1.970 [ABC (G)]Complexity = 57.998 + 14.646 [ABC (G)]MV = 72.605 + 5.014 [ABC (G)](2)
Regression models forRA (G)] are as follows:Polarity = 8.149 + 1.483 [RA (G)]Refractive index = 20.601 + 3.739 [RA (G)]Complexity = 21.629 + 27.488 [RA (G)]MV = 62.686 + 9.01 [RA (G)](3)
Regression models for sumS (G) are as follows:Polarity = 6.574 + 1.536 [S (G)]Refractive index = 16.611 + 3.874 [S (G)]Complexity = 2.985 + 27.955 [S (G)]MV = 58.881 + 9.330 [S (G)](4)
Regression models forGA (G) are as follows:Polarity = 8.253 + 0.683 [GA (G)]Refractive index = 20.859 + 1.721 [GA (G)]Complexity = −1.382 + 13.231 [GA (G)]MV = 66.421 + 4.207 [GA (G)](5)
Regression models forM1 (G) are as follows:Polarity = 14.306 + 0.090 [M1 (G)]Refractive index = 36.118 + 0.227 [M1 (G)]Complexity = 104.000 + 1.804 [M1 (G)]MV = 93.419 + 0.597 [M1 (G)](6)
Regression models forHM (G) are as follows:Polarity = 19.733 + 0.023 [HM (G)]Refractive index = 49.809 + 0.030 [HM (G)]Complexity = 187.760 + 0.255 [HM (G)]MV = 122.143 + 0.084 [HM (G)](7)
Regression models forM2 (G) are as follows:Polarity = 18.556 + 0.057 [M2 (G)]Refractive index = 46.844 + 0.143 [M2 (G)]Complexity = 158.746 + 1.228 [M2 (G)]MV = 116.976 + 0.390 [M2 (G)](8)
Regression models forF (G) are as follows:Polarity = 19.971 + 0.021 [F (G)]Refractive index = 50.401 + 0.054 [F (G)]Complexity = 205.354 + 0.442 [F (G)]MV = 122.674 + 0.151 [F (G)](9)
Regression models forH (G) are as follows:Polarity = 6.602 + 1.34 [H (G)]Refractive index = 16.703 + 4.373 [H (G)]Complexity = −14.384 + 32.514 [H (G)]MV = 58.494 + 10.565 [H (G)]


### 3.2. Quantitative Structure Analysis and Comparison between Topological Indices and Correlation Coefficient of Physicochemical Properties

Table3 consists of physicochemical properties related to ten rheumatoid arthritis drugs. Their TIs values, on the other hand, are recorded in Table 1 and derived with the aid of their molecular structure. Table 2 shows the correlation coefficients among physicochemical properties and TIs. Figure 2 depicts a graph of the correlation coefficient of aforementioned drugs.Table 3
RA immune system drugs physical properties.
Name of drugMolar volume (cm3)Polarity (m3mol−1)Refractive index (m3mol−1)ComplexityBoiling point (°C)D (cm3)Enthalpy (°C)Azatjoprine145.4027.3068.90354.00555.80145.40368.50Hydroxycholoqoine285.4039.2099.00331.00516.70285.40266.30Sulfasatazne267.7040.60102.40657.00689.30267.70370.70Filgotinib281.1045.30114.30715.00281.10Leflumomide194.1024.2061.00327.00289.30194.10129.00Predisolane274.7037.9095.50724.00570.60274.70314.80Methotrexate275.7047.20119.00704.00275.70Bracitinib238.1038.9098.20678.00707.00238.10Tofacitinib241.0034.7087.50585.80241.00308.10Upadacinib243.0036.3091.60561.00189.00243.00Figure 2
Physicochemical properties and TIs.
(a)(b)(c)(d)


### 3.3. Calculation of Statistical Parameters

In this section, QSPR modelling is imposed in such a way to determine a relationship between the physicochemical properties of rheumatoid arthritis drugs that are tofacitinib, leflunomide, upadacinib, baricitinib, filgotinib, methotrexate, neoral, prednisolone, sulfasalazine, azathioprine, and hydroxychloroquine. TIs,b, r, and N are independent variables, regression model constants, correlation coefficients, and sample size. Such type of test is useful for comparing and deciding on model improvements. It is worth noting that the r value is larger than 0.6 and the p value is less than 0.05. As a result, it determines that all properties are significant as given in Tables 4–12.Table 4
QSPR model ofABC (G).
Physiochemical propertyNAbrr2FpIndicatorPolarity1010.3640.80.7810.87225.4790.001SignificantRefractive index1026.1741.9700.7820.76025.3640.001SignificantMolar volume1072.6055.0140.8980.80633.1570.000SignificantComplexity957.99814.6460.7090.5027.0640.033SignificantTable 5
QSPR model ofRA (G).
Physiochemical propertyNAbRr2FpIndicatorPolarity108.1491.4830.9090.82738.1290.000SignificantRefractive index1020.6013.7390.9090.82537.8290.000SignificantMolar volume1062.6869.3010.9140.83540.5110.000SignificantComplexity921.62927.4880.7290.5327.9590.026SignificantTable 6
QSPR model ofS (G).
Physiochemical propertyNABRr2FpIndicatorPolarity106.5741.5360.9330.87153.9790.000SignificantRefractive index1016.6113.8740.9330.87053.5950.000SignificantMolar volume1058.8819.3300.9080.82537.7240.000SignificantComplexity92.98527.9550.7350.5408.2330.024SignificantTable 7
QSPR model ofGA (G).
Physiochemical propertyNAbRr2FpIndicatorPolarity108.2530.6830.9070.82337.2920.000SignificantRefractive index1020.8591.7210.9070.82237.0200.000SignificantMolar volume1066.4214.2070.8970.80432.7700.000SignificantComplexity9−1.38213.2310.7610.5809.6500.017SignificantTable 8
QSPR model ofM1 (G).
Physiochemical propertyNAbRr2FpIndicatorPolarity1014.3060.0900.8180.66916.1880.004SignificantRefractive index1036.1180.2270.8180.66816.1300.004SignificantMolar volume1093.4190.5970.8670.75324.3280.001SignificantComplexity9104.0001.8040.7090.5037.0730.032SignificantTable 9
QSPR model ofM2 (G).
Physiochemical propertyNAbRr2FpIndicatorPolarity1018.5560.0570.7540.56910.5700.012SignificantRefractive index1046.8440.1430.7540.56810.5300.012SignificantMolar volume10116.9760.3900.8300.68817.6690.003SignificantComplexity9158.7461.2280.7070.5006.9880.033SignificantTable 10
QSPR model ofHM (G).
Physiochemical propertyNAbRr2FpIndicatorPolarity1019.7330.0230.7280.5309.0090.017SignificantRefractive index1049.8090.0300.7270.5298.9790.017SignificantMolar volume10122.1430.0840.8200.67216.3940.004SignificantComplexity9187.7600.2550.6770.4585.9090.045SignificantTable 11
QSPR model ofF (G).
Physiochemical propertyNAbRr2FpIndicatorPolarity1019.9710.0210.7220.5218.6950.018SignificantRefractive index1050.4010.0540.7210.5208.6730.019SignificantMolar volume10122.6740.1510.8210.67316.4910.004SignificantComplexity9205.3540.4420.6490.4215.0970.059SignificantTable 12
QSPR model ofH (G).
Physiochemical propertyNAbRr2FpIndicatorPolarity106.6021.340.9310.86752.1340.000SignificantRefractive index1016.7034.3730.9300.86654.5650.000SignificantMolar volume1058.49410.5650.9090.82638.0350.000SignificantComplexity9−14.38432.5140.7550.5699.2570.019Significant


### 3.4. Standard Error of Estimate (SE) and Comparision

A standard error of estimate is the measure of variation for an observation calculated around the computed regression line. This assesses degree of correctness of predictions computed around regression line as shown in Table13. Tables 14–17 compare experimental and theoretical calculated values of the models.Table 13
SE of estimate.
Topological indexStd. error of the estimate forPolarityRefractive indexComplexityMolar volumeABC (G)3.701099.35334131.7552220.82685RA (G)3.153067.98066127.7541619.18354S (G)2.720186.88389126.5986519.75959GA (G)3.182078.05203121.0912520.92556M1 (G)4.3512810.9985131.7140523.49938M2 (G)4.9695512.55069132.112726.37188HM (G)54.925813.11165137.5209627.05248F (G)5.2411213.23146142.039326.9989H (G)2.761587.00027122.5470119.69254Table 14
Comparison of actual and computed values for refractive index from regression models.
Name of drugRefractivity of drug (m3·mol−1)Refractivity from regression model forABC (G)R (G)S (G)GA (G)M1 (G)M2 (G)F (G)H (G)HM (G)Azathioprine68.9 ± 0.566.637865.46964.842366.2589868.80671.72672.54165.4619572.549Hydroxycholoquine99 ± 0.3100.8764100.653100.250798.4244799.22497.32399.86599.2652498.469Sulfasalazine102.4 ± 0.587.815391.4176691.5341690.731686.05885.88384.20593.667884.969Filgotinib114.3 ± 0.5109.7217108.5423111.0591110.7985109.212107.762107.101108.8858106.869Leflunomide61 ± 0.368.824567.712466.2756866.981870.62273.15674.91766.6863974.469Prednisolone95 ± 0.4109.8202108.804110.0519109.5421112.39113.911113.905107.7051113.229Methotrexate119 ± 0.3109.2883113.7021114.5845111.8655104.218100.18399.325116.013899.369Baricitinib98.2 ± 0.593.390492.4271993.4711693.6917294.2394.60694.89792.4870994.569Tofacitinib87.5 ± 0.392.7691.4176691.572991.2134893.32293.31995.11390.3880594.149Upadacitinib91.6 ± 0.598.394296.6148797.8875297.8565498.7798.46799.10996.0292298.529Table 15
Comparision of actual and computed values for polarity from regression models.
Name of drugPolarity of drug (cm3)Polarity computed from regression model forABC (G)R (G)S (G)GA (G)M1 (G)M2 (G)F (G)H (G)HM (G)Azathioprine27.326.4057425.94525.697226.2705427.26628.47428.58121.54337.167Hydroxycholoquine39.239.9795239.9000339.736239.0358139.32638.67739.20731.901257.039Sulfasalazine40.634.8014936.2370236.280235.982834.10634.11733.11730.18646.689Filgotinib45.343.4862143.0291644.021743.9465843.28642.83842.02134.849263.479Leflunomide24.227.2726526.834826.265526.557427.98629.04429.50521.918238.639Prednisolone37.943.5252643.1329743.622343.4479944.54645.28944.66734.487468.355Methotrexate47.243.3143945.075745.419444.3700441.30639.81738.99737.033457.729Baricitinib38.937.0117236.6374337.048237.1575637.34637.59437.27529.824254.049Tofacitinib34.736.761836.2370236.295636.1740436.98637.08137.35929.18153.727Upadacitinib36.338.9954638.2983938.799338.8104239.14639.13338.91330.909657.085Table 16
Comparison of actual and computed values for complexity from regression models.
Name of drugComplexity of drugComplexity computed from regression model forABC (G)R (G)S (G)GA (G)M1 (G)M2 (G)F (G)H (G)HM (G)Azathioprine354358.8268351.485351.0248347.6518363.776372.418386.574348.1471381.05Hydroxycholoquine331613.3743610.1471606.5335594.9392605.512592.23610.226599.4803601.37Sulfasalazine657516.2713542.2517543.6347535.7966500.88493.99482.046557.8624486.62Filgotinib715679.1349668.1468684.5279690.0701684.888681.874669.454671.0111672.77Leflunomide327375.0839367.9778361.3681353.2088378.208384.698406.022357.251397.37Prednisolone724679.8672670.0709677.2596680.4114710.144734.678725.146662.2323726.83Methotrexate704675.9127706.0802709.967698.2733645.2616.79605.806724.0089609.02Baricitinib678557.7195549.6735557.6122558.5539565.824568.898569.562549.0836568.22Tofacitinib553.0328542.2517543.9143539.5013558.608557.846571.33533.4769564.65Upadacitinib561594.9204580.46589.4809590.5729601.904602.054604.038575.42601.88Table 17
Comparison of actual and computed values for molar volume from regression models.
Name of drugMolar volume molar volume of drug (cm3)Molar volume from regression model forABC (G)R (G)S (G)GA (G)M1 (G)M2 (G)F (G)H (G)HM (G)Azathioprine145.4 ± 7.0175.5926174.298175.0395177.4017179.387184.836184.58477.1802185.815Hydroxycholoquine285.4 ± 3.0262.7359261.8204260.3157256.0305259.385254.646260.9995.94864258.391Sulfasalazine267.7 ± 3.0229.4931238.8469239.3232237.2252224.759223.446217.292.8408220.591Filgotinib281.1 ± 7.0285.2487281.4455286.3464286.2788285.653283.116281.224101.2902281.911Leflunomide194.1 ± 3.0181.1581179.8786178.4916179.1686184.163188.736191.22877.86004191.191Prednisolone274.7 ± 5.0285.4994282.0966283.9206283.2077294.011299.886300.25100.6347299.719Methotrexate275.7 ± 3.0284.1457294.2809294.8367288.8872272.519262.446259.48105.2479260.911Baricitinib238.1 ± 7.0243.6827241.3582243.9882244.4612246.251247.236247.09892.18524247.471Tofacitinib241 ± 3.0242.0782238.8469239.4165238.4032243.863243.726247.70291.0198246.295Upadacitinib243 ± 7.0256.4182251.7753254.6244254.6422258.191257.766258.87694.15192258.559


## 3.1. Regression Model

In this article, the tenacity of QSPR modelling is tested using a drug computable structure analysis of nine topological indices. Table2 shows the four physical properties of ten RA medications: polarity, molar volume (MV), refractive index (R), and complexity. We run the regression analysis for the drugs, and the validated linear regression model is as follows:(9)P=A+bTI.Table 2
Correlation coefficients.
Topological indexCorrelation coefficient ofPolarityRefractive indexMolar volumeComplexityABC (G)0.8720.8720.8980.709RA (G)0.9090.9090.9140.729S (G)0.9330.9330.9080.735GA (G)0.9070.9070.8970.761M1 (G)0.8180.8180.8670.709M2 (G)0.7540.7540.830.707HM (G)0.7280.7270.820.677F (G)0.7220.7210.8210.649H (G)0.9310.930.9090.755P denotes physicochemical property. Letters A, TI, and b represent the topological index, constant, and regression coefficient, respectively. The software packages Statistix, Python, and MATLAB are helpful in determining the results. The physiochemical properties of nine TIs of RA drugs are investigated using a linear QSPR model. For the aforementioned calculation, equation (1) is appropriate.Theorem 1.
LetG1 be a graph of Sulfasatazne. The TIs of G1 are(10)ABCG1=31.29,RAG1=18.95,SG1=19.34,GAG1=40.60,M1G1=220,M2G1=273,FG1=626,HG1=17.60,HMG1=1172.Proof.
LetG1 be the graph of sulfasatazne, with edge set E and Em,n partition of G1 vertices with degrees m and n. E1,2=2,, ,E1,3=13,E1,4=2,E2,3=6,E3,3=18, and E3,4=2.(i)
Using Definition 1 and the partitionsEm,n, we obtain(11)ABCG1=21+2−21×2+131+3−21×3+21+4−21×4+62+3−22×3+183+3−23×3+23+4−23×4=31.29.(ii)
Using Definition 2 and the partitionsEm,n, we obtain(12)RAG1=211×2+1311×3+211×4+612×3+1813×3+213×4=18.95.(iii)
Using Definition 3 and the partitionsEm,n, we obtain(13)SG1=211+2+1311+3+211+4+612+3+1813+3+213+4=19.34.(iv)
Using Definition 4 and the partitionsEm,n, we obtain(14)GAG1=21×21+2+131×31+3+21×41+4+62×32+3+183×33+3+23×43+4=40.60.(v)
Using Definition 5 and the partitionsEm,n, we obtain(15)M1G1=21+2+131+3+21+4+62+3+183+3+23+4=220.(iii)
Using Definition 5 and the partitionsEm,n, we obtain(16)M2G1=21×2+131×3+21×4+62×3+183×3+23×4=273.(vi)
Using Definition 6 and the partitionsEm,n, we obtain(17)HG1=211+2+1311+3+211+4+612+3+1813+3+213+4=17.60.(vii)
Using Definition 7 and the partitionsEm,n, we obtain(18)HMG1=21+22+131+32+21+42+62+32+182+42+23+42=1172.(viii)
Using Definition 8 and the partitionsEm,n, we obtain(19)FG1=21+4+131+9+21+16+64+9+184+16+29+9+29+16=626.Theorem 2.
LetG2 be graph of Upadacinib. The TIs of G2 are as follows:(20)ABCG2=36.66,RAG2=20.33,SG2=20.98,GAG2=44.74,M1G2=276,M2G2=361,FG2=902,HG2=18.14,HMG2=1624.Proof.
LetG2 be graph of upadacinib having edge set E and Em,n partition of G2 vertices with degrees m and n. With E1,3=7,E1,4=16,E2,3=4,E3,3=12,E3,4=4, and E4,4=6.(i)
Using Definition1 and the partitions Em,n, we obtain(21)ABCG2=71+3−21×3+161+4−21×4+42+3−22×3+123+3−23×3+43+4−23×4+614×4=36.66.(ii)
Using Definition2 and the partitions Em,n, we obtain(22)RAG2=711×3+1611×4+412×3+1213×3+413×4+614×4=20.33.(iii)
Using Definition3 and the partitions Em,n, we obtain(23)SG2=711+3+1611+4+412+3+1213+3+413+4+614+4=20.98.(iv)
Using Definition4 and the partitions Em,n, we obtain(24)GAG2=71×31+3+161×41+4+42×32+3+123×33+3+43×43+4+64×44+4=44.74.(v)
Using Definition5 and the partitions Em,n, we obtain(25)M1G2=∑uv∈EG1su+sv=71+3+161+4+42+3+123+3+43+4+64+4=276.(vi)
Using Definition5 and the partitions Em,n, we obtain(26)M2G2=71×3+161×4+42×3+123×3+43×4+64×4=361.(vii)
Using Definition5 and the partitions Em,n, we obtain(27)HG2=711+3+1611+4+412+3+1213+3+413+4+614+4=18.14.(viii)
Using Definition6 and the partitions Em,n, we obtain(28)HMG2=31+22+31+32+61+42+42+32+52+42+33+32+13+42+44+42=1624.(ix)
Using Definition7 and the partitions Em,n, we obtain(29)FG2=31+4+41+9+61+16+44+9+54+16+39+9+19+16+416+16=902.
The remaining drugs topological indices can be calculated using the same procedure as applied in Theorem2, Theorem 1, and Definitions 1–8. Table 1 contains values for each drug.
Applying (1), we calculated the linear models for all TIs written as follows:(1)
Regression models forABC (G) are as follows:Polarity = 10.364 + 0.781 [ABC (G)]Refractive index = 26.174 + 1.970 [ABC (G)]Complexity = 57.998 + 14.646 [ABC (G)]MV = 72.605 + 5.014 [ABC (G)](2)
Regression models forRA (G)] are as follows:Polarity = 8.149 + 1.483 [RA (G)]Refractive index = 20.601 + 3.739 [RA (G)]Complexity = 21.629 + 27.488 [RA (G)]MV = 62.686 + 9.01 [RA (G)](3)
Regression models for sumS (G) are as follows:Polarity = 6.574 + 1.536 [S (G)]Refractive index = 16.611 + 3.874 [S (G)]Complexity = 2.985 + 27.955 [S (G)]MV = 58.881 + 9.330 [S (G)](4)
Regression models forGA (G) are as follows:Polarity = 8.253 + 0.683 [GA (G)]Refractive index = 20.859 + 1.721 [GA (G)]Complexity = −1.382 + 13.231 [GA (G)]MV = 66.421 + 4.207 [GA (G)](5)
Regression models forM1 (G) are as follows:Polarity = 14.306 + 0.090 [M1 (G)]Refractive index = 36.118 + 0.227 [M1 (G)]Complexity = 104.000 + 1.804 [M1 (G)]MV = 93.419 + 0.597 [M1 (G)](6)
Regression models forHM (G) are as follows:Polarity = 19.733 + 0.023 [HM (G)]Refractive index = 49.809 + 0.030 [HM (G)]Complexity = 187.760 + 0.255 [HM (G)]MV = 122.143 + 0.084 [HM (G)](7)
Regression models forM2 (G) are as follows:Polarity = 18.556 + 0.057 [M2 (G)]Refractive index = 46.844 + 0.143 [M2 (G)]Complexity = 158.746 + 1.228 [M2 (G)]MV = 116.976 + 0.390 [M2 (G)](8)
Regression models forF (G) are as follows:Polarity = 19.971 + 0.021 [F (G)]Refractive index = 50.401 + 0.054 [F (G)]Complexity = 205.354 + 0.442 [F (G)]MV = 122.674 + 0.151 [F (G)](9)
Regression models forH (G) are as follows:Polarity = 6.602 + 1.34 [H (G)]Refractive index = 16.703 + 4.373 [H (G)]Complexity = −14.384 + 32.514 [H (G)]MV = 58.494 + 10.565 [H (G)]


## 3.2. Quantitative Structure Analysis and Comparison between Topological Indices and Correlation Coefficient of Physicochemical Properties

Table3 consists of physicochemical properties related to ten rheumatoid arthritis drugs. Their TIs values, on the other hand, are recorded in Table 1 and derived with the aid of their molecular structure. Table 2 shows the correlation coefficients among physicochemical properties and TIs. Figure 2 depicts a graph of the correlation coefficient of aforementioned drugs.Table 3
RA immune system drugs physical properties.
Name of drugMolar volume (cm3)Polarity (m3mol−1)Refractive index (m3mol−1)ComplexityBoiling point (°C)D (cm3)Enthalpy (°C)Azatjoprine145.4027.3068.90354.00555.80145.40368.50Hydroxycholoqoine285.4039.2099.00331.00516.70285.40266.30Sulfasatazne267.7040.60102.40657.00689.30267.70370.70Filgotinib281.1045.30114.30715.00281.10Leflumomide194.1024.2061.00327.00289.30194.10129.00Predisolane274.7037.9095.50724.00570.60274.70314.80Methotrexate275.7047.20119.00704.00275.70Bracitinib238.1038.9098.20678.00707.00238.10Tofacitinib241.0034.7087.50585.80241.00308.10Upadacinib243.0036.3091.60561.00189.00243.00Figure 2
Physicochemical properties and TIs.
(a)(b)(c)(d)


## 3.3. Calculation of Statistical Parameters

In this section, QSPR modelling is imposed in such a way to determine a relationship between the physicochemical properties of rheumatoid arthritis drugs that are tofacitinib, leflunomide, upadacinib, baricitinib, filgotinib, methotrexate, neoral, prednisolone, sulfasalazine, azathioprine, and hydroxychloroquine. TIs,b, r, and N are independent variables, regression model constants, correlation coefficients, and sample size. Such type of test is useful for comparing and deciding on model improvements. It is worth noting that the r value is larger than 0.6 and the p value is less than 0.05. As a result, it determines that all properties are significant as given in Tables 4–12.Table 4
QSPR model ofABC (G).
Physiochemical propertyNAbrr2FpIndicatorPolarity1010.3640.80.7810.87225.4790.001SignificantRefractive index1026.1741.9700.7820.76025.3640.001SignificantMolar volume1072.6055.0140.8980.80633.1570.000SignificantComplexity957.99814.6460.7090.5027.0640.033SignificantTable 5
QSPR model ofRA (G).
Physiochemical propertyNAbRr2FpIndicatorPolarity108.1491.4830.9090.82738.1290.000SignificantRefractive index1020.6013.7390.9090.82537.8290.000SignificantMolar volume1062.6869.3010.9140.83540.5110.000SignificantComplexity921.62927.4880.7290.5327.9590.026SignificantTable 6
QSPR model ofS (G).
Physiochemical propertyNABRr2FpIndicatorPolarity106.5741.5360.9330.87153.9790.000SignificantRefractive index1016.6113.8740.9330.87053.5950.000SignificantMolar volume1058.8819.3300.9080.82537.7240.000SignificantComplexity92.98527.9550.7350.5408.2330.024SignificantTable 7
QSPR model ofGA (G).
Physiochemical propertyNAbRr2FpIndicatorPolarity108.2530.6830.9070.82337.2920.000SignificantRefractive index1020.8591.7210.9070.82237.0200.000SignificantMolar volume1066.4214.2070.8970.80432.7700.000SignificantComplexity9−1.38213.2310.7610.5809.6500.017SignificantTable 8
QSPR model ofM1 (G).
Physiochemical propertyNAbRr2FpIndicatorPolarity1014.3060.0900.8180.66916.1880.004SignificantRefractive index1036.1180.2270.8180.66816.1300.004SignificantMolar volume1093.4190.5970.8670.75324.3280.001SignificantComplexity9104.0001.8040.7090.5037.0730.032SignificantTable 9
QSPR model ofM2 (G).
Physiochemical propertyNAbRr2FpIndicatorPolarity1018.5560.0570.7540.56910.5700.012SignificantRefractive index1046.8440.1430.7540.56810.5300.012SignificantMolar volume10116.9760.3900.8300.68817.6690.003SignificantComplexity9158.7461.2280.7070.5006.9880.033SignificantTable 10
QSPR model ofHM (G).
Physiochemical propertyNAbRr2FpIndicatorPolarity1019.7330.0230.7280.5309.0090.017SignificantRefractive index1049.8090.0300.7270.5298.9790.017SignificantMolar volume10122.1430.0840.8200.67216.3940.004SignificantComplexity9187.7600.2550.6770.4585.9090.045SignificantTable 11
QSPR model ofF (G).
Physiochemical propertyNAbRr2FpIndicatorPolarity1019.9710.0210.7220.5218.6950.018SignificantRefractive index1050.4010.0540.7210.5208.6730.019SignificantMolar volume10122.6740.1510.8210.67316.4910.004SignificantComplexity9205.3540.4420.6490.4215.0970.059SignificantTable 12
QSPR model ofH (G).
Physiochemical propertyNAbRr2FpIndicatorPolarity106.6021.340.9310.86752.1340.000SignificantRefractive index1016.7034.3730.9300.86654.5650.000SignificantMolar volume1058.49410.5650.9090.82638.0350.000SignificantComplexity9−14.38432.5140.7550.5699.2570.019Significant


## 3.4. Standard Error of Estimate (SE) and Comparision

A standard error of estimate is the measure of variation for an observation calculated around the computed regression line. This assesses degree of correctness of predictions computed around regression line as shown in Table13. Tables 14–17 compare experimental and theoretical calculated values of the models.Table 13
SE of estimate.
Topological indexStd. error of the estimate forPolarityRefractive indexComplexityMolar volumeABC (G)3.701099.35334131.7552220.82685RA (G)3.153067.98066127.7541619.18354S (G)2.720186.88389126.5986519.75959GA (G)3.182078.05203121.0912520.92556M1 (G)4.3512810.9985131.7140523.49938M2 (G)4.9695512.55069132.112726.37188HM (G)54.925813.11165137.5209627.05248F (G)5.2411213.23146142.039326.9989H (G)2.761587.00027122.5470119.69254Table 14
Comparison of actual and computed values for refractive index from regression models.
Name of drugRefractivity of drug (m3·mol−1)Refractivity from regression model forABC (G)R (G)S (G)GA (G)M1 (G)M2 (G)F (G)H (G)HM (G)Azathioprine68.9 ± 0.566.637865.46964.842366.2589868.80671.72672.54165.4619572.549Hydroxycholoquine99 ± 0.3100.8764100.653100.250798.4244799.22497.32399.86599.2652498.469Sulfasalazine102.4 ± 0.587.815391.4176691.5341690.731686.05885.88384.20593.667884.969Filgotinib114.3 ± 0.5109.7217108.5423111.0591110.7985109.212107.762107.101108.8858106.869Leflunomide61 ± 0.368.824567.712466.2756866.981870.62273.15674.91766.6863974.469Prednisolone95 ± 0.4109.8202108.804110.0519109.5421112.39113.911113.905107.7051113.229Methotrexate119 ± 0.3109.2883113.7021114.5845111.8655104.218100.18399.325116.013899.369Baricitinib98.2 ± 0.593.390492.4271993.4711693.6917294.2394.60694.89792.4870994.569Tofacitinib87.5 ± 0.392.7691.4176691.572991.2134893.32293.31995.11390.3880594.149Upadacitinib91.6 ± 0.598.394296.6148797.8875297.8565498.7798.46799.10996.0292298.529Table 15
Comparision of actual and computed values for polarity from regression models.
Name of drugPolarity of drug (cm3)Polarity computed from regression model forABC (G)R (G)S (G)GA (G)M1 (G)M2 (G)F (G)H (G)HM (G)Azathioprine27.326.4057425.94525.697226.2705427.26628.47428.58121.54337.167Hydroxycholoquine39.239.9795239.9000339.736239.0358139.32638.67739.20731.901257.039Sulfasalazine40.634.8014936.2370236.280235.982834.10634.11733.11730.18646.689Filgotinib45.343.4862143.0291644.021743.9465843.28642.83842.02134.849263.479Leflunomide24.227.2726526.834826.265526.557427.98629.04429.50521.918238.639Prednisolone37.943.5252643.1329743.622343.4479944.54645.28944.66734.487468.355Methotrexate47.243.3143945.075745.419444.3700441.30639.81738.99737.033457.729Baricitinib38.937.0117236.6374337.048237.1575637.34637.59437.27529.824254.049Tofacitinib34.736.761836.2370236.295636.1740436.98637.08137.35929.18153.727Upadacitinib36.338.9954638.2983938.799338.8104239.14639.13338.91330.909657.085Table 16
Comparison of actual and computed values for complexity from regression models.
Name of drugComplexity of drugComplexity computed from regression model forABC (G)R (G)S (G)GA (G)M1 (G)M2 (G)F (G)H (G)HM (G)Azathioprine354358.8268351.485351.0248347.6518363.776372.418386.574348.1471381.05Hydroxycholoquine331613.3743610.1471606.5335594.9392605.512592.23610.226599.4803601.37Sulfasalazine657516.2713542.2517543.6347535.7966500.88493.99482.046557.8624486.62Filgotinib715679.1349668.1468684.5279690.0701684.888681.874669.454671.0111672.77Leflunomide327375.0839367.9778361.3681353.2088378.208384.698406.022357.251397.37Prednisolone724679.8672670.0709677.2596680.4114710.144734.678725.146662.2323726.83Methotrexate704675.9127706.0802709.967698.2733645.2616.79605.806724.0089609.02Baricitinib678557.7195549.6735557.6122558.5539565.824568.898569.562549.0836568.22Tofacitinib553.0328542.2517543.9143539.5013558.608557.846571.33533.4769564.65Upadacitinib561594.9204580.46589.4809590.5729601.904602.054604.038575.42601.88Table 17
Comparison of actual and computed values for molar volume from regression models.
Name of drugMolar volume molar volume of drug (cm3)Molar volume from regression model forABC (G)R (G)S (G)GA (G)M1 (G)M2 (G)F (G)H (G)HM (G)Azathioprine145.4 ± 7.0175.5926174.298175.0395177.4017179.387184.836184.58477.1802185.815Hydroxycholoquine285.4 ± 3.0262.7359261.8204260.3157256.0305259.385254.646260.9995.94864258.391Sulfasalazine267.7 ± 3.0229.4931238.8469239.3232237.2252224.759223.446217.292.8408220.591Filgotinib281.1 ± 7.0285.2487281.4455286.3464286.2788285.653283.116281.224101.2902281.911Leflunomide194.1 ± 3.0181.1581179.8786178.4916179.1686184.163188.736191.22877.86004191.191Prednisolone274.7 ± 5.0285.4994282.0966283.9206283.2077294.011299.886300.25100.6347299.719Methotrexate275.7 ± 3.0284.1457294.2809294.8367288.8872272.519262.446259.48105.2479260.911Baricitinib238.1 ± 7.0243.6827241.3582243.9882244.4612246.251247.236247.09892.18524247.471Tofacitinib241 ± 3.0242.0782238.8469239.4165238.4032243.863243.726247.70291.0198246.295Upadacitinib243 ± 7.0256.4182251.7753254.6244254.6422258.191257.766258.87694.15192258.559


## 4. Conclusions

According to the statistical parameters used in linear QSPR models and topological indices, theS (G) index has a high correlation with molar volume (r = 0.908). The H index has the highest correlated complexity value, r = 0.755. The S (G) index represents the highest correlation coefficient of the refractive index, r = 0.933. The maximum correlated value of flash polarity r = 0.933 is provided by harmonic S (G). TIs had no relationship with density, boiling point, or polar surface area.In this paper, we calculated TIs and linked it with the linear QSPR model for rheumatoid arthritis drugs. The findings will aid in the development of new drugs and preventive measures for numerous disease in the pharmaceutical manufacturing industries. The correlation coefficient contributes significantly to the range of TIs for drugs. These findings are eye-opening of pharmaceutical researchers who work on medication science, and they provide a method to estimate and predict physicochemical properties for new RA treatment drugs to cure other specific autoimmune diseases.



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*Source: 1562125-2022-08-05.xml*