After the preparation of the drug solution, UV scanning was conducted over the wavelength range of 200–400 nm. The spectra of all three drugs exhibited absorbance within the challenging range of 240–280 nm, as depicted in the zero-order spectrum (Fig. 2). To distinguish individual drugs within the ternary mixture, specific methods were applied to each D0 spectra individually. This step is crucial for accurate identification and quantification of the individual components in the complex mixture. According to the reported method, the UV spectrophotometric methods can be divided into different windows as the methods selected in the determination of three drugs fall in Window 1 (W1), which utilizes zero-order absorption spectra, and Window 3 (W3), which is based on ratio spectra.

Fig. 2

Overlay of three drugs of UV spectra, ACE (10 µg/mL)—red, PAR (32.5 µg/mL)—blue, and TRM (3.75 µg/mL)—green

Method 1: double divisor ratio spectra method (DDRSM)Determination of ACE

In this method, the first step involves generation of double divisor by adding the standard spectra of PAR′ and TRM′ to obtain a PAR′ TRM′ followed by a constant generation, where the standard spectra of PAR and TRM are divided by the double divisor PAR′ TRM′. The resulting ternary mixture spectrum is divided by the double divisor to obtain a spectrum, which is then subtracted from the constant and subsequently multiplied by the double divisor PAR′ TRM′ to yield a D0 spectra of ACE at a wavelength of 276 nm. The overall separation process of ACE from the ternary mixture with the help of spectrums is depicted in Fig. 3

Fig. 3

The process of how the ACE (10 µg/mL) was separated from the ternary mixture

Determination of PAR

To determine the D0 spectra of PAR, the double devisor was prepared by adding the standard spectra of ACE′ with TRM′ to yield ACE′ TRM′ and then constant is prepared by dividing the standard spectra of ACE and TRM by the double divisor (ACE′ TRM′). The mixture spectrum is then divided by the double divisor, subtracted from the constant, and multiplied by the double divisor to obtain a D0 spectra of PAR at a wavelength of 247 nm. The overall separation process of ACE from the ternary mixture with the help of spectrums is depicted in Fig. 4.

Fig. 4

The process of how the PAR (32.5 µg/mL) was separated from the ternary mixture

Determination of TRM

This is achieved using spectra manager software. The above-obtained spectra of ACE and PAR are added together which are subtracted from ternary mixture spectra to obtain D0 spectra of TRM at a wavelength of 270 nm. The overall separation process of ACE from the ternary mixture with the help of spectrums is depicted in Fig. 5.

Fig. 5

The process of how the TRM (3.75 µg/mL) was separated from the ternary mixture

Method 2: Area under the curve method (AUC)Determination of ACE

AUC is calculated for the D0 spectra obtained from the method 1 to determine the concentration of the separated spectra. This can be performed by selecting a wavelength range of approximately 205–225 nm for the D0 spectra (Fig. 6a).

Fig. 6

Area under the curve selection for the three drugs a ACE (10 µg/mL), b PAR (32.5 µg/mL), and c TRM (3.75 µg/mL) at different wavelength ranges

Determination of PAR

AUC is calculated by selecting a wavelength range of approximately 235–255 nm for the AUC spectra (Fig. 6b) which is obtained in the above method. Furthermore, the spectra were utilized to calculate the marketed formulation unknown concentration.

Determination of TRM

AUC is determined for the obtained TRM spectra by selecting a wavelength range of approximately 260–280 nm (Fig. 6c).

Method validation

In accordance with ICH recommendations Q2R1, the developed methods underwent thorough validation, and specific parameters were assessed, including linearity, limit of quantification, detection, accuracy, and precision [30, 35,36,37].

Precision

The ACE, PAR, and TRM weights were measured precisely. Six repetitions of the identical process were carried out to ensure method repeatability, accounting for both intra- and interday fluctuations. Analyzing the drug solution three times in a single day allowed for the determination of intraday precision. By examining solutions with the same concentrations on three distinct days in a week, interday precision is determined in Table 1.

Table 1 Results for ACE, PAR, and TRM in terms of accuracy and precisionAccuracy

Applying the standard addition method to a medication sample containing known concentrations of ACE, PAR, and TRM standard drugs is equal to 50%, 100%, and 150% of the label claim allowed for the assessment of the method's accuracy. The experiment's results are listed in Table 1.

Stability

Working solutions of ACE, PAR, and TRM were stored in tightly sealed containers at approximately 4 °C for three weeks. Volumetric flasks containing these solutions were covered with aluminum foil to ensure protection.

Linearity

The method’s linearity was determined by evaluating five different concentrations ranging from ACE (8–12 µg/mL), PAR (22.75–35.75 µg/mL), and TRM (2.62–4.12 µg/mL), respectively. The linearity spectra for each separated drugs are depicted in Fig. 7, and the results are depicted in Table 2. The detection limit (LOD) and quantification limit (LOQ) for ACE, PAR, and TRM were estimated using the calibration curve approach. LOD represents the smallest detectable amount of an analyte, while LOQ is the smallest quantifiable amount with appropriate precision and accuracy. The results are shown in Table 2.

Fig. 7

Linearity spectra for the three drugs a ACE (8–12 µg/mL), b PAR (22.75–35.75 µg/mL), c TRM (2.62–4.12 µg/mL) at different concentration ranges

Table 2 Linearity data for the proposed methodAssay of marketed formulation

The proposed spectrophotometric techniques were applied to analyze a commercially available tablet, Zerodol-PT. The mean drug content ranged from 99.5 ± 1.25. No interference peaks were observed in the spectra, indicating accurate approximation of the medication without considering excipients, and the results are depicted in Table 3.

Table 3 Assay of tablet formulationStatistical analysis

A statistical comparison between the proposed method and reported method [38] is suggested for ACE, PAR, and TRM analysis in pharmaceutical dosage form with respect to assay, and the results are depicted in Table 4. The calculated Student’s t and F test with a null hypothesis (H0) stating no difference observed between the values and an alternative hypothesis (Ha) stating the maximum difference between the values is observed. The values obtained were found to be less than the tabulated ones.

Table 4 Determination by Student t test and F test for the proposed method and reported method [38]Green assessment for the proposed method

The development of the method was rooted in adherence to the twelve principles of analytical green chemistry. Initially, the solvent selection was done based on the G score; it was found that ethanol has a G score of 6.6 which is near to water (7.3). To assert the eco-friendliness of an analytical approach, substantiation through assessment tools is imperative. In this case, two distinct assessment tools were employed for evaluating the developed method. The first, AES, underwent manual computation for its assessment. Conversely, the GAPI assessment utilized software termed the GAPI chart generator version 1.0 which was employed for crafting the GAPI chart, while AGREE metric calculations were conducted utilizing AGREE calculator version 0.5 beta. This comprehensive approach ensured a thorough evaluation of the method's environmental sustainability [39,40,41,42,43,44]

Green analytical procedure index (GAPI)

GAPI stands as a qualitative gauge utilized to appraise the ecological soundness of an analytical methodology. Operating as a semi-quantitative instrument, it employs pictorial depictions to gauge the environmental sustainability of analytical procedures. By integrating eco-friendly ethanol solvents into the methodology, a method was developed to symbolize a green emblem, highlighting its environmentally mindful characteristics. The GAPI assessment incorporates a color-coded representation of the GAPI value within the final pictogram, offering a visual indicator of its eco-friendliness. Comprising 15 discernible stages, the GAPI evaluation framework is meticulously structured within the GAPI software interface. The proposed method underwent a comprehensive assessment via the GAPI software, with the outcomes elucidated in Table 5 showcasing its environmental viability.

Table 5 Comparison of green metric tools between developed method and proposed methodsAnalytical eco-scale (AES)

The AES assessment relies on the assignment of penalty points (PP) to chemicals involved. PP is derived through a graphical representation illustrating the chemicals employed in the process. It encompasses four primary evaluation stages, culminating in the AES computation formula: AES = 100—PP.

In the initial stage, ethanol is depicted with three pictograms, resulting in an overall PP of 3. Proceeding to the second stage, as the quantity of solvent or reagent utilized per sample is less than 10 mL, the PP is calculated as 3 × 2 (pictogram score), yielding 6.

Moving to the third stage, with the UV energy consumption per analysis falling below 0.1 kWh, the PP is set at 0. Subsequently, in the fourth stage, the method’s waste of solvents, known to have environmental ramifications, is evaluated based on the employed wastage recycling approach, leading to a PP of 0.

Consequently, with the cumulative PP loss for the devised technique totaling 9, the resulting AES score for the developed method stands at 94.

Analytic GREEnness (AGREE)

AGREE metrics is an innovative tool crafted to gauge how environmentally friendly. It provides a detailed look at how these methods impact the environment. The results from AGREE are displayed as a circular graph split into 12 sections, each reflecting one of the twelve green analytical chemistry principles. Every section gets a score between 0 and 1, where 1 signifies the highest level of eco-friendliness. The average score appears at the center of the graph, and the closer it is to 1, the better it is for the environment. The software is designed to fully integrate the established methodology, ensuring that the outcomes reflect the most environmentally friendly approach possible. You can see a summary of the AGREE findings in Table 5. Application of the present method to the AGREE tool yielded a score of 0.91, indicating its strong alignment with GAC principles and highlighting its environmentally friendly analytical attributes.

The developed method underwent a thorough comparison with established HPLC techniques to evaluate its efficiency. The term “greenness” denotes not only the absence of efficiency drawbacks in the developed method but also its exceptional ecological safety profile. When scrutinized for environmental impact alongside other methods, the developed approach exhibited significantly higher greenness scores [45]. These findings, including the comparative analysis results, are detailed in Table 5.

*ACE and PAR dissolve readily in organic solvents and completely insoluble in water. Based on this aspect, any method which utilized water as a solvent to dissolve these drugs was not considered into account because it is practically not possible [47]. Pharmaceutical formulations typically use organic solvents such as ethanol, methanol, chloroform, and acetone to dissolve active ingredients. Both compounds are more soluble in organic solvents than in water; however, this might vary depending on the solvent and other conditions such as temperature [48].