# Application of Rank Annihilation Factor Analysis for Antibacterial Drugs Determination by Means of pH Gradual Change-UV Spectral Data

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{−1}for both components). The limits of detection were 0.25 and 0.38 µg mL

^{−1}for SMX and TMP, respectively. The proposed method was successfully applied to the simultaneous determination of SMX and TMP in some synthetic, pharmaceutical formulation and biological fluid samples. In addition, the means of the estimated RSD (%) were 1.71 and 2.18 for SMX and TMP, respectively, in synthetic mixtures. The accuracy of the proposed method was confirmed by spiked recovery test on biological samples with satisfactory results (90.50–109.80%).

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Instruments

#### 2.2. Chemicals and Reagents

#### 2.3. Standard Solutions

^{−1}) were separately prepared by dissolving 10.0 mg of each standard powder in the least amount of acetonitrile and completed to volume with deionized water. Suitable amounts of primary stock solutions of SMX and TMP were transferred to 25.0 mL volumetric flasks followed by the addition of 5.0 mL of the Britton–Robinson buffer solution (0.1 mol/L) and diluted with water to prepare standard working solutions (1, 5, 10, 15, 20, 25 and 30 µg mL

^{−1}for both SMX and TMP). All solutions were stored at 4 °C and equilibrated to room temperature before use. The working standard solutions for spike recovery were also prepared at required concentrations (10, 20 and 30 µg mL

^{−1}for SMX and 10, 20 and 25 µg mL

^{−1}for TMP) from the stock standard solutions.

#### 2.4. Treatment of Real Samples

^{−1}) and TMP (10, 20 and 25 µg mL

^{−1}). Then, 10.0 mL of methanol was mixed properly with the solution for protein precipitation. The solution was subjected to a centrifugal force of 4000 rpm for 10 min, in order to separate the precipitated proteins. The process continued by passing the clear supernatant layer through a 0.45 µm Millipore filter. The percolated solution was collected and transferred into a volumetric flask of 25.0 mL followed by adding 10 mL of Britton–Robinson buffer solution (with considered pH) and completed to volume with deionized water and then the resulting solution was introduced to the spectrophotometer cell.

#### 2.5. Spectrophotometric Analysis of Sulfamethoxazole and Trimethoprim in Laboratory-Prepared Mixtures

^{−1}for both SMX and TMP. The absorbance spectra, from 200 to 350 nm, of these laboratory-prepared mixtures were recorded. Concentrations of the analytes in the prepared samples were estimated from the corresponding RAFA models using standard solutions of 5.0 µg mL

^{−1}SMX and 5.0 µg mL

^{−1}TMP.

#### 2.6. Theory of Rank Annihilation Factor Analysis

**R**indicates the residual matrix,

**S**the single component standard matrix and

**M**, the mixture matrix, the following equation can be written:

**R**is one less than of

**M**, by which the contribution of the single component should be removed from matrix

**M**. Hence, the concentration of the component in a mixture can be estimated using the following equation:

_{s}and C

_{x}are the component concentrations in standard and mixture solutions, respectively. To select the parameter k properly, principal component analysis (PCA) was applied iteratively to the residual matrix (

**R**) resulting in different values of k on each run, for which the RSD (Relative Standard Deviation) was assessed afterward. The RSD is an assessment of principal component model fit deficiency for a dataset. As far as the residual matrix RSD value reaches its minimum, decomposition of matrix

**R**(Equation (3)) is required to achieve the optimal solution. The RSD is calculated as stated below [35]:

_{i}represents the eigenvalue, n is the number of principal components and c is the rank of the data matrix.

#### 2.7. Chemometrics Models

^{−1}for SMX and 1.0–30.0 µg mL

^{−1}for TMP. The concentrations of all considered levels for each compound are based on the design requirements. All mixtures of this design were used as a validation set to test the predictive capability of the developed multivariate RAFA model that was built by using standard solutions of 5.0 µg mL

^{−1}SMX and TMP. All spectra of mixtures were mean-centered, in preparation for modeling. RAFA was carried out using a laboratory-created program, routinely implemented in MATLAB 6.5.

## 3. Results and Discussion

#### 3.1. Linear Calibration Models for A Single Component

_{max}(SMX, 254.5 nm; TMP, 282 nm) vs. sample concentration gives the linear range for the considered component. Table 1 illustrates the calibration models and the respective figures of merit. Linear dynamic ranges (LDRs) were 1.0–30.0 µg mL

^{−1}for both compounds and the coefficients of determination (R

^{2}) were 0.994 and 0.996 for SMX and TMP, respectively. Detection limits were achieved equal to 0.25 and 0.38 µg mL

^{−1}for SMX and TMP, which are proper values for analysis of drugs. In this research, performing a five-level full factorial design (comprising 25 solutions) made it possible to select a set of mixtures covering the entire experimental domain. The contribution of components was assured to be additive and in accordance to the linear range of the spectrophotometer. Table 2 shows the actual and predicted concentrations of SMX and TMP in synthetic mixtures. As can be seen, the accuracy of the results is satisfactory in all cases, when the concentration ratio of SMX and TMP vary from 1:30 to 30:1. The RSD values are all <4.0%, which shows the reproducibility of the method.

#### 3.2. Selection of the pH Range

_{2}-NH- group. Trimethoprim is a basic compound and a proton is associated with the NH

_{2}substituents in acidic solution, but the bathochromic shift occurs as the pH is increased [36]. Additionally, there is no significant variation between the spectra recorded at pH 8 to 12 for SMX. For TMP, however, no considerable change was observed in the pH range 9 to 12. Therefore, it seems that there is no considerable information in pH region 9–12, thus the range 2–9 (including 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5 and 9.0) was selected for the determination of SMX and TMP.

#### 3.3. pH-Spectral Absorbance Data: Bilinearity, Trilinearity and Rank Deficiency

**D**, made up of n columns (301 wavelength points) and m rows (15 pH points), representing the number of wavelengths and pH values, respectively. The following equation describes the application of Lambert–Beer law to obtain the matrix

**D**:

**D**(m × n),

**C**(m × r) and

**S**(n × r), and the superscript T is an indication of a transposed matrix. Matrix

**S**comprises particular spectra of the different chemical forms of SMX (or TMP), while matrix

**C**integrates the concentrations of these forms at different pH values. Consequently, matrix

**D**, with n rows and p columns, is a bilinear pH–spectra matrix.

^{−1}. Considering the shape of the spectra of the two analytes in different concentrations (from 1 to 30 µg mL

^{−1}), it seems that altering the concentrations of SMX and TMP results in small changes below 220 nm and above 320 nm. Therefore, to meet the trilinear conditions, only the spectral interval between 220 and 320 nm was selected for further chemometric analysis.

#### 3.4. Rank Analysis

_{a}values of these two compounds, besides the different forms of the species in acidic and basic solutions, are shown in Figure 4 and Figure 5.

_{tot}= [SMX

^{+}] + [SMX] + [SMX

^{−}].

SMX + H

_{2}O ⇆ SMX

^{−}+ H

_{3}O

^{+}

SMX

^{+}+ H

_{2}O ⇆ SMX + H

_{3}O

^{+}

^{−}is an SMX molecule which has lost its proton, SMX

^{+}is an SMX molecule which has gain a proton and SMX represents a neutral SMX molecule.

_{a}value of about 7 has been taken into account [46,47], while TMP actually has two pK

_{a}values, pK

_{a1}and pK

_{a2}, which are about 1.32 and 7.45, respectively. Secondly, different descriptions of the TMP protonation process in acidic pH ranges makes this compound more challenging to observe. The protonation of TMP is demonstrated as proton absorption by an amino group in some references (e.g., [48]); nonetheless, the study of the results obtained by NMR spectroscopy [49] and capillary zone electrophoresis [50] define it as a two-step process involving two heterocyclic nitrogen atoms (N1 and N3) (see Figure 5).

_{a2}values between 6 and 7 and full protonation at pH 2.1. As for most nitrogen heterocycles, pK

_{a1}values are likely to be near 1 or 2 [52], and protonation of TMP is supposed to occur at the N3 position [51].

^{2+}+ H

_{2}O ⇆ TMP

^{+}+ H

_{3}O

^{+}pK

_{a1}= 1.35

TMP

^{+}+ H

_{2}O ⇆ TMP + H

_{3}O

^{+}pK

_{a2}= 7.45

^{+}is a TMP molecule which has gain a proton, TMP

^{2+}is a TMP molecule which has gain two protons and TMP represents a TMP neutral molecule)

_{tot}= [TMP

^{2+}] + [TMP

^{+}] + [TMP], the concentration of different TMP forms at various pH values can be obtained. The results of these computations are shown in Figure 4. Thus, at pH values lower than pK

_{a1}, both nitrogen rings would be protonated (TMP

^{2+}), while, at the pH range between pK

_{a1}to pK

_{a2}, a proton is being released to form the TMP

^{+}. Finally, at pH values above pK

_{a2}, TMP loses the second proton and the compound becomes neutral (TMP).

#### 3.5. Determination of SMX and TMP in Validation Samples

^{−1}standard single-solute solution of SMX (or TMP) for building the model, multicomponent solutions with a wide range of SMX and TMP concentrations were also made to evaluate the performance of the model. As mentioned in Section 3.2, calibration was carried out deploying three PCs to build the model for both SMX and TMP determination.

**S**and

**M**, which represent the calibration standard set and the sample set, respectively. As discussed in Section 3, the first step is to estimate the rank of

**R**, which equals

**M**-k.

**S**, using SVD. Then, an iterative procedure, plotting the eigenvalues (or singular values) of least significant PCs of

**R**, is applied as a function of k to find the minimum value of k. Figure 5c gives an example of finding minimum k-value for determining the concentration of the analyte in the mixture, here with the k-value 1.21. The concentration of the intended analyte in the calibration standard was 5.0 µg mL

^{−1}; therefore, the anticipated concentration of this analyte in the unknown sample was determined as 6.05 µg mL

^{−1}.

#### 3.6. Determination of SMX and TMP in Real Samples

## 4. Conclusions

^{−1}for both components). The presented method was also capable of a precise determination of SMX and TMP in pharmaceutical formulations and biological samples with high recoveries. The principal advantages of this approach over other techniques are: (1) the complicated operation of pre-separation can be omitted; (2) it is easy to produce second-order data using this approach; and (3) the required equipment in this technique is widely available. We may also take advantage of this method for SMX and TMP determination in other types of samples, such as urine and wastewater. However, the main discussion of this paper only concerns the method application for determination of SMX and TMP in some biological and pharmaceutical samples.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**UV absorbance spectra of sulfamethoxazole (5.0 µg mL

^{−1}) and trimethoprim (5.0 µg mL

^{−1}) at different pHs (pH from 2 to 12 at intervals of 0.5).

**Figure 2.**UV absorbance spectra of (

**a**) sulfamethoxazole (pH = 5.4) at different concentrations (1–30 µg mL

^{−1}); and (

**b**) trimethoprim (pH = 7.0) at different concentrations (1–30 µg mL

^{−1}).

**Figure 3.**Eigenvalue vs. row number in Evolving Factor Analysis to determine the number of species contributing to the spectral signal of sulfamethoxazole and trimethoprim standard sample solutions.

**Figure 5.**The structures of sulfamethoxazole (

**a**) and trimethoprim (

**b**) at different pH values; and (

**c**) RSD values obtained for parameter k values in RAFA analysis.

Parameters | Sulfamethoxazole | Trimethoprim |
---|---|---|

Linear range (µg mL^{−1}) | 1.0–30.0 | 1.0–30.0 |

Correlation coefficient | 0.994 | 0.996 |

Intercept | −0.0196 ± 0.0033 (n = 3) | −0.0121 ± 0.0025 (n = 3) |

Slope (mL^{−1} µg) | 0.0632 ± 0.0028 (n = 3 ) | 0.0197 ± 0.0020 (n = 3) |

Detection limit (µg mL^{−1}) | 0.25 | 0.38 |

**Table 2.**The concentrations of SMX and TMP and the results of the replicate measurements for determination of SMX and TMP in synthetic mixtures.

Sample No. | SMX | TMP | ||||||
---|---|---|---|---|---|---|---|---|

Actual (µg mL ^{−1}) | Predicted (µg mL ^{−1}) | Error (µg mL ^{−1}) | RSD (%, n = 3) | Actual (µg mL ^{−1}) | Predicted (µg mL ^{−1}) | Error (µg mL ^{−1}) | RSD (%, n = 3) | |

1 | 1 | 0.91 | −0.09 | 3.1 | 1 | 1.03 | 0.03 | 2.78 |

2 | 6 | 5.65 | −0.35 | 2.32 | 1 | 0.98 | −0.02 | 2.12 |

3 | 13 | 14.59 | 1.59 | 1.51 | 1 | 0.95 | −0.05 | 2.98 |

4 | 20 | 20.18 | 0.18 | 3.20 | 1 | 1.08 | 0.08 | 3.15 |

5 | 30 | 30.55 | 0.55 | 0.55 | 1 | 0.96 | −0.04 | 3.19 |

6 | 1 | 0.89 | −0.11 | 1.68 | 6 | 5.66 | −0.34 | 1.59 |

7 | 6 | 6.58 | 0.58 | 2.32 | 6 | 6.12 | 0.12 | 2.31 |

8 | 13 | 12.77 | −0.23 | 1.98 | 6 | 6.23 | 0.23 | 1.66 |

9 | 20 | 19.32 | −0.68 | 1.33 | 6 | 5.89 | −0.11 | 1.45 |

10 | 30 | 28.51 | −1.49 | 2.52 | 6 | 6.08 | 0.08 | 1.98 |

11 | 1 | 1.05 | 0.05 | 2.31 | 13 | 13.78 | 0.78 | 1.32 |

12 | 6 | 6.87 | 0.87 | 1.56 | 13 | 12.32 | −0.68 | 1.75 |

13 | 13 | 13.88 | 0.88 | 1.23 | 13 | 12.42 | −0.58 | 2.13 |

14 | 20 | 21.89 | 1.89 | 1.32 | 13 | 13.88 | 0.88 | 1.55 |

15 | 30 | 29.12 | −0.88 | 1.45 | 13 | 13.59 | 0.59 | 1.32 |

16 | 1 | 1.10 | 0.1 | 1.05 | 20 | 20.13 | 0.13 | 2.62 |

17 | 6 | 5.64 | −0.36 | 1.23 | 20 | 21.16 | 1.16 | 1.78 |

18 | 13 | 14.02 | 1.02 | 2.12 | 20 | 19.55 | −0.45 | 3.21 |

19 | 20 | 22.21 | 2.21 | 1.21 | 20 | 19.17 | −0.83 | 2.17 |

20 | 30 | 28.65 | −1.35 | 1.59 | 20 | 21.21 | 1.21 | 2.79 |

21 | 1 | 0.99 | −0.01 | 2.32 | 30 | 32.42 | 2.42 | 1.98 |

22 | 6 | 5.22 | −0.78 | 1.48 | 30 | 30.67 | 0.67 | 3.31 |

23 | 13 | 14.65 | 1.65 | 1.75 | 30 | 29.10 | −0.9 | 2.78 |

24 | 20 | 20.03 | 0.03 | 0.75 | 30 | 31.78 | 1.78 | 1.44 |

25 | 30 | 31.22 | 1.22 | 0.89 | 30 | 32.52 | 2.52 | 1.32 |

**Table 3.**Results of the replicate measurements (n = 3) for the determination of SMX and TMP in some pharmaceutical formulations.

Drug | SMX | TMP | ||||
---|---|---|---|---|---|---|

Approximate Doses (mg) | Proposed Method (mg) | HPLC (mg) | Approximate Doses (mg) | Proposed Method (mg) | HPLC (mg) | |

Co-trimoxazole adult tablet | 400 | 403 ± 4.3 | 402 ± 5.2 | 80 | 78.3 ± 2.6 | 79.6 ± 3.1 |

Co-trimoxazole pediatric tablet | 100 | 98.23 ± 3.7 | 97.3 ± 4.1 | 20 | 22.3 ± 3.1 | 24.8 ± 4.1 |

Co-trimoxazole oral suspension | 200 | 197.56 ± 4.9 | 198.4 ± 4.7 | 40 | 38.5 ± 4.5 | 38.1 ± 4.3 |

Co-trimoxazole intravenous infusion | 400 | 398.91 ± 4.2 | 400.1 ± 5.1 | 80 | 82.8 ± 5.2 | 81.6 ± 6.1 |

**Table 4.**Results of the replicate measurements (n = 3) for the determination of SMX and TMP in biological fluids.

SMX | ||||||||

Samples | Proposed Method | HPLC | ||||||

Added(µg mL^{−1}) | Found(µg mL^{−1}) | Recovery(%) | RSD(%, n = 3) | Added(µg mL^{−1}) | Found(µg mL^{−1}) | Recovery(%) | RSD(%, n = 3) | |

Serum | 10 | 10.98 | 109.8 | 5.54 | 10 | 9.3 | 93.0 | 6.5 |

20 | 19.10 | 95.5 | 6.32 | 20 | 19.05 | 95.2 | 6.95 | |

30 | 32.31 | 107.7 | 3.52 | 30 | 33.05 | 110.1 | 5.32 | |

Plasma | 10 | 9.23 | 92.3 | 4.21 | 10 | 9.35 | 93.5 | 5.62 |

20 | 21.56 | 107.8 | 3.11 | 20 | 19.32 | 96.6 | 4.73 | |

30 | 29.11 | 97.0 | 4.63 | 30 | 32.15 | 107.2 | 6.72 | |

TMP | ||||||||

Samples | Proposed Method | HPLC | ||||||

Added(µg mL^{−1}) | Found(µg mL^{−1}) | Recovery(%) | RSD(%, n = 3) | Added(µg mL^{−1}) | Found(µg mL^{−1}) | Recovery(%) | RSD(%, n = 3) | |

Serum | 20 | 20.89 | 104.4 | 4.62 | 20 | 21.65 | 108.2 | 5.97 |

10 | 9.05 | 90.5 | 5.31 | 10 | 10.21 | 102.1 | 4.78 | |

25 | 24.32 | 97.3 | 4.65 | 25 | 23.65 | 94.6 | 5.64 | |

Plasma | 20 | 19.52 | 97.6 | 3.12 | 20 | 19.23 | 96.15 | 4.68 |

10 | 10.23 | 102.3 | 2.65 | 10 | 9.53 | 95.3 | 6.28 | |

25 | 26.14 | 104.6 | 4.77 | 25 | 27.15 | 108.6 | 6.89 |

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**MDPI and ACS Style**

Esteki, M.; Dashtaki, E.; Heyden, Y.V.; Simal-Gandara, J.
Application of Rank Annihilation Factor Analysis for Antibacterial Drugs Determination by Means of pH Gradual Change-UV Spectral Data. *Antibiotics* **2020**, *9*, 383.
https://doi.org/10.3390/antibiotics9070383

**AMA Style**

Esteki M, Dashtaki E, Heyden YV, Simal-Gandara J.
Application of Rank Annihilation Factor Analysis for Antibacterial Drugs Determination by Means of pH Gradual Change-UV Spectral Data. *Antibiotics*. 2020; 9(7):383.
https://doi.org/10.3390/antibiotics9070383

**Chicago/Turabian Style**

Esteki, Mahnaz, Elham Dashtaki, Yvan Vander Heyden, and Jesus Simal-Gandara.
2020. "Application of Rank Annihilation Factor Analysis for Antibacterial Drugs Determination by Means of pH Gradual Change-UV Spectral Data" *Antibiotics* 9, no. 7: 383.
https://doi.org/10.3390/antibiotics9070383