The main goal of this study was to establish a low-cost, simple, sensitive and accurate analytical method for the simultaneous determination of SMX and TMP in their mixtures, drug formulations and biological fluids, as well as to construct a model with satisfactory accuracy for effective analytical practice.
3.1. Linear Calibration Models for A Single Component
Calibration curves were plotted to find the linear dynamic range of each component. The absorbance spectra were recorded in the range 200–350 nm relative to a solvent blank. Plotting the absorbance at its λmax
(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 (R2
) 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.
The absorbance spectra of the two analytes recorded in the range of 200–350 nm are shown in Figure 1
and Figure 2
. A strong spectral overlap is observed, which complicates the individual determination of the compounds from the spectrum of a mixture.
3.2. Selection of the pH Range
shows the spectra of SMX and TMP in different pH buffer solutions (pH 2–12). By increasing pH from 2 to 12 at intervals of 0.5, a hypsochromic shift takes place for SMX, whereas a bathochromic one can be observed for TMP. These results are in agreement with a previous report by Zhou & Moore [36
]. Sulfamethoxazole is an acidic compound and its spectrum undergoes a hypsochromic shift with increasing pH, which is related to the loss of a proton from the -SO2
-NH- group. Trimethoprim is a basic compound and a proton is associated with the NH2
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
To perform multi-component calibration, RAFA utilizes two data matrices simultaneously; the first matrix is composed of the unknowns and the second of calibration samples. The rank annihilation technique would entail linear and additive measured signals; the constructed data in this way are called bilinear. Rank annihilation requires the signal for the intended analyte to be identical for all samples, as well as to be independent of the remaining substituents signal [37
]. The data collected from a given concentration of SMX (and/or TMP) are composed of absorbance spectra at consecutive pH values, with the absorbance values recorded at n wavelength points (200–350 nm at intervals of 0.5 nm, 301 points) in each spectrum. Then, it is possible to arrange the spectra of the solution at a variety of pH values to form a data matrix 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
where the dimensions of these matrices are 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.
As discussed above, various types of second-order data constructed by different methods, can be calibrated through second-order calibration techniques to quantify target analyte(s) accurately, even if there are unmodeled or unexpected interferents in the mixture [26
]. To obtain a robust calibration model and take advantage of the second order, a true trilinear dataset should be utilized as starting point [26
]. If the pure analyte response shows the same form in both standard and unknown mixtures, then the joint variation in the standard and unknown mixtures can be modeled using trilinear models [39
Regarding the chemical aspects, the concentration–absorbance–pH data that are produced for acid/base species necessarily should be trilinear. It should be mentioned that second-order calibration models would be considerably limited, if applied to the data deviating slightly from trilinearity [40
]. In most cases, the primary reason for such deviations are changes in the peak shapes [40
]. Figure 2
shows the spectral profile of SMX and TMP in their linear concentration range from 1 to 30 µg mL−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.
Capitalizing on the analyte’s acidic/basic characteristics, we can choose the pH as parameter to modulate the second data dimension. In these situations, the generated data are linearly dependent on the pH profiles and the concentrations of proton-transfer species are correlated [41
]. When more than one analyte is present, the aforementioned dependency gives rise to rank-deficiency, which means that the comprehensive rank of the measured data and the sum of the ranks of the individual species contribution do not yield the same value [41
]. This problem is observed under conditions that the number of independent reactions is smaller than the number of response-active species, i.e. when the number of independent components is lower than the number of real chemical components present in the system. The rank deficiency is definitely one of the significant sources of deviation from trilinearity. However, rank deficiency can be eliminated deploying a matrix augmentation method, which is the simultaneous analysis of the corresponding matrix along with one additional full rank standard matrix. Some authors [42
] have discussed the primary concept of the augmentation effect on solving the rank deficiency. Based on this explanation and regarding the abovementioned conditions in which the problem has to be solved, rank deficiency removal was performed using column-wise augmentation together with a standard data matrix. Because of this method implementation, an improved resolution of the system would be achieved.
3.4. Rank Analysis
In this study, Evolving Factor Analysis was used to determine the number of species contributing to the spectral signal [43
]. This method offers the opportunity to determine the number of chemical components, through investigating the eigenvalues of the submatrices, resulting from spectroscopic data that are arranged in ascending order of the evolutionary variable (pH). Considering only the initiating spectrum, adding successive spectra to the previous sub-matrix gives rise to a succession of sub-matrices, of which the eigenvalues are extracted, normalized to unit sum, and finally their logarithms are plotted as a function of the sequence number. Visual inspection of the plot gives an indication of real factor appearance above the noise level. Based on this method, by employing the singular value decomposition (SVD) algorithm, it is possible to achieve the matrix rank by calculating the eigenvalues of the first row, then the first and the second row, and so on until all rows. Finally, the eigenvalues that are obtained in each step are plotted. The number of results higher than the noise level implies the rank of the matrix. Figure 3
illustrates the eigenvalues regarding each row of the given data matrices and shows the rank of the single-solute solutions of both SMX and TMP equals 2. This can be justified regarding existing species in the standard solution of SMX and TMP, at different pH values. The pKa
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
As mentioned by Babić and Chen [44
], observing previously performed studies, an agreement about the SMX molecular forms at different pH values exists. As depicted in Figure 5
, SMX has a sulfonamide group with pKa value of 5.6 and an amine attached to the aromatic ring, with a pKa of 1.7 (referred to as N1 and N4, respectively). Between 1.7 and 5.6, the molecules are not neutral. One could state that, to have only neutral SMX for the amine group, pH should be above 4.7 (pKa + 3). However, at this pH, about 10% of the sulfonamide group is already negatively charged. The same reasoning is valid for the sulfonamide group (uncharged below 2.6 (pKa − 3), but then 10% of amine groups is already charged. The equilibrium concentrations of different forms of SMX at various pH values can be achieved using the following equilibrium relations, considering that
[SMX]tot = [SMX+] + [SMX] + [SMX−].
SMX + H2O ⇆ SMX− + H3O+
SMX+ + H2O ⇆ SMX + H3O+
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.
depicts the presence of the three absorbing compounds in the pH range of 2–12 that we considered in this work.
The discussion about TMP, however, is somehow controversial. Firstly, in many references, only a single pKa
value of about 7 has been taken into account [46
], while TMP actually has two pKa
, 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
The hydrogen bonding analysis and molecular packing of TMP and other comparable compounds, using computational methods and crystal structure determinations, were carried out [51
]. Others have claimed that the favored site for protonation is on the N1 (ring nitrogen) [50
], leading to pKa2
values between 6 and 7 and full protonation at pH 2.1. As for most nitrogen heterocycles, pKa1
values are likely to be near 1 or 2 [52
], and protonation of TMP is supposed to occur at the N3 position [51
Accordingly, at different pH values, the following equilibriums are established:
TMP2+ + H2O ⇆ TMP+ + H3O+ pKa1 = 1.35
TMP+ + H2O ⇆ TMP + H3O+ pKa2 = 7.45
is a TMP molecule which has gain a proton, TMP2+
is a TMP molecule which has gain two protons and TMP represents a TMP neutral molecule)
Based on these equilibrium equations and considering that [TMP]tot
] + [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 pKa1
, both nitrogen rings would be protonated (TMP2+
), while, at the pH range between pKa1
, a proton is being released to form the TMP+
. Finally, at pH values above pKa2
, TMP loses the second proton and the compound becomes neutral (TMP).
Spectrophotometric analysis based on pH modulation provides rank-deficient data matrices for mixture of components with acid–base behavior [53
]. If the number of significant factors contributing to the data variances, which is determined through singular value decomposition or other factor analysis approaches, is lower than the real number of existing chemical components in the signal contribution, the data matrix is supposed to be rank deficient. For closed-reaction systems, such as those studied here, the total concentration of the mixture is constant during pH alteration for each of the acid–base pair components, which implies mixtures with a lower number of independent reactions than response-active absorbing species.
Therefore, it is clear why the graphs in Figure 3
determine rank two for a standard solution including SMX (or TMP), under conditions in which there are three absorbing species in the current system. When changing the pH is used for modulating the second data dimension, concentrations exhibit linear dependencies on the pH profiles, which is the consistency of the aggregate individual proton-transferring species. For a system containing more than one analyte, this dependency would lead to rank-deficiency, which means that the overall rank of the measured data is not equal to the sum of the individual species contribution ranks [42
]. Therefore, it is obvious that the rank of a mixture including SMX and TMP is determined as 3, whereas the individual summation of rank contribution of SMX and TMP equals 4.
3.5. Determination of SMX and TMP in Validation Samples
Given a spectrum, which is the result of different components absorbance, rank annihilation factor analysis (RAFA) gives us the opportunity to quantify a specific component contributing to the spectrum, while the quantification and identification of the other components is not necessary. The performance of RAFA in SMX and TMP determination, in both validation samples and pharmaceuticals, was carried out. The possibility of uncalibrated and unexpected interferents existence in the samples is also noteworthy. In addition to the preparation of a 5 µg mL−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.
A two-way data matrix was constructed by measuring the absorbance under different conditions in terms of wavelength and pH values, while the concentration of the analyte was kept constant.
As mentioned above, the RAFA methodology entails two bilinear datasets, S
, 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
, 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 5
c 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
The analysis of the synthetic mixtures was carried out for the measurement of the prediction error, which brought about the results illustrated in Table 1
and Table 2
, representing individual errors less than 5.6% and 3.0% for the calculated overall prediction error.