Holistically benchmarked optimization-assisted green spectrophotometric–chemometric approach for simultaneous determination of piroxicam and venlafaxine in pharmaceutical formulations and spiked human plasma

Abstract

The recent clinical introduction of the piroxicam (PIR) and venlafaxine hydrochloride (VEN) combination has highlighted the need for analytical methodologies capable of simultaneously quantifying both drugs while minimizing the environmental burden of routine pharmaceutical analysis. To date, only a chromatographic method has been reported for this purpose, which is associated with relatively high organic solvent consumption, greater instrumental complexity, and increased energy requirements. A green UV spectrophotometric platform integrated with chemometric modeling was therefore established for the concurrent determination of PIR and VEN in pharmaceutical formulations and as a proof-of-concept application to spiked human plasma. Spectral measurements were performed using a water–ethanol (1:1, v/v) solvent system, whereas methanol was employed solely for plasma protein precipitation. Experimental design efficiency was enhanced by combining Brereton’s multilevel calibration design with a Maximin Distance Design (MMD) for validation sample selection, enabling representative calibration and validation datasets to be generated with a reduced number of experimental mixtures. Severe spectral overlap between the two analytes was effectively resolved using six well-established chemometric approaches: CLS, PCR, PLS, GA-PLS, FA-PLS, and MCR-ALS. Among the investigated models, MCR-ALS showed the best overall numerical performance, achieving correlation coefficients of up to 0.9998 for pharmaceutical formulations and 0.9996 for spiked human plasma. The corresponding limits of detection were 0.125 and 0.029 µg mL⁻¹ for VEN and PIR, respectively, in pharmaceutical formulations, and 0.245 and 0.058 µg mL⁻¹ in spiked human plasma. Across the investigated concentration ranges, the proposed method also exhibited satisfactory accuracy and precision. The sustainability performance of the analytical platform was evaluated using multiple complementary metrics, with the pharmaceutical assay compared with the previously reported chromatographic procedure according to the evaluated criteria. The proposed pharmaceutical spectrophotometric assay showed a more favorable sustainability profile according to the evaluated criteria, including environmental and practical applicability dimensions. The application to spiked human plasma provided a proof-of-concept assessment of matrix applicability. Nevertheless, additional investigations involving incurred clinical specimens and full bioanalytical validation are required before the methodology can be considered suitable for routine clinical practice or pharmacokinetic studies.

Supplementary Information

The online version contains supplementary material available at https://doi.org/10.1186/s13065-026-01929-4.

Introduction

The transition toward more sustainable pharmaceutical practices has accelerated the development of analytical methodologies that deliver reliable analytical performance while minimizing environmental impact, simplifying analytical procedures, and improving accessibility [1–5]. Throughout the pharmaceutical lifecycle, analytical chemistry is indispensable for drug discovery, formulation development, quality assurance, and environmental surveillance. Nevertheless, many conventional analytical techniques remain dependent on large volumes of organic solvents, sophisticated instrumentation, and energy-intensive workflows, leading to increased operational costs, chemical waste, and environmental burden. These challenges have driven the evolution of Green Analytical Chemistry (GAC) into a broader sustainability-oriented framework that integrates environmental responsibility with analytical quality, efficient resource utilization, operational feasibility, and societal benefit [6, 7].

Piroxicam (PIR) (Fig. S1), a potent nonsteroidal anti-inflammatory drug, is extensively prescribed for the management of inflammatory and musculoskeletal disorders owing to its prolonged half-life and strong analgesic activity [8]. Venlafaxine hydrochloride (VEN) (Fig. S1), a serotonin–norepinephrine reuptake inhibitor, is widely utilized for the treatment of depression, anxiety disorders, and neuropathic pain [9, 10]. Recently, the combined administration of PIR and VEN has attracted growing pharmaceutical and clinical interest, particularly in pain-associated depressive conditions requiring simultaneous anti-inflammatory and antidepressant therapy [11, 12]. However, increasing reports concerning the misuse and abuse potential of VEN-containing formulations have raised additional analytical and forensic concerns, necessitating reliable methodologies capable of sensitive and selective determination of both drugs in pharmaceutical formulations and biological matrices [13, 14].

A review of the available literature indicates that only one analytical procedure has been described for the simultaneous determination of the PIR–VEN combination [15]. This reported method employs reversed-phase liquid chromatography and provides excellent analytical performance; however, it depends on relatively high consumption of organic solvents, dedicated chromatographic instrumentation, and continuous energy input. Such requirements may restrict its routine application, particularly in laboratories with limited analytical infrastructure. Consequently, there remains a clear need for alternative methodologies that maintain satisfactory analytical performance while reducing environmental impact, instrumental demands, and operational complexity.

In parallel with method development, contemporary sustainable analytical chemistry recognizes that the environmental merit of an analytical procedure should be assessed through comprehensive benchmarking rather than a single greenness indicator [16–18]. Such an approach enables objective verification of genuine sustainability gains rather than incremental methodological modification [19, 20]. Employing multiple complementary sustainability assessment tools enables a more balanced evaluation by simultaneously addressing environmental performance, practical utility, innovation, and overall sustainability, thereby minimizing potential bias associated with reliance on a single metric.

Among environmentally benign analytical techniques, UV spectrophotometry combined with multivariate chemometric analysis offers an attractive solution for resolving mixtures with severely overlapping spectra. This strategy couples simple and widely available instrumentation with the capability for simultaneous multicomponent quantification while substantially decreasing solvent consumption relative to chromatographic methods [21]. Moreover, the incorporation of rational experimental design approaches, including Brereton’s multilevel calibration design and Maximin Distance Design (MMD) for validation sample selection, facilitates the construction of representative calibration and validation datasets using fewer experimental mixtures [22, 23], thereby improving experimental efficiency and conserving reagents without compromising predictive performance [24, 25].

It is important to note that the novelty of the present work does not lie in the development of new chemometric algorithms or mathematical formulations. Instead, the study combines well-established multivariate calibration models with rational experimental design principles and comprehensive sustainability benchmarking within a unified analytical framework for the simultaneous determination of PIR and VEN.

Accordingly, a green UV spectrophotometric–chemometric platform was established for the concurrent determination of PIR and VEN in pharmaceutical formulations and spiked human plasma using a water–ethanol (1:1, v/v) solvent system for spectral measurements. The analytical performance of six established chemometric models were comparatively investigated, while Brereton’s multilevel calibration design and MMD-based validation sample selection were implemented to enhance experimental efficiency. The proposed methodology was benchmarked against the previously reported chromatographic procedure using complementary sustainability assessment tools encompassing environmental, practical, innovation, and holistic sustainability dimensions. In addition, its applicability to biologically relevant matrices was demonstrated through the analysis of spiked human plasma samples, providing a proof-of-concept assessment of the developed platform under biological conditions.

Experimental

Instrumentation and computational environment

A UV-1800 double-beam UV–Visible spectrophotometer (Shimadzu Corporation, Kyoto, Japan) with matched 1.0 cm quartz cells (Hellma Analytics, Müllheim, Germany) was used to measure UV absorption spectra. Under regulated laboratory circumstances (25 ± 0.5 °C), spectral acquisition was conducted with a 1.0 nm bandwidth, 0.5 nm wavelength increment, and medium scanning speed. UVProbe software (Version 2.62, Shimadzu Corporation, Kyoto, Japan) controlled instrument operation, data acquisition, baseline correction, and spectral analysis.

An AUX220 analytical balance (Shimadzu Corporation, Kyoto, Japan) with 0.1 mg readability weighed samples. A 40 kHz USC300TH ultrasonic bath (Julabo GmbH, Seelbach, Germany) with digital temperature regulation helped dissolve samples.

Chemometric and statistical computations were done in MATLAB R2023a (MathWorks Inc., Natick, MA, USA). PLS_Toolbox Version 9.1 (Eigenvector Research Inc., Manson, WA, USA) was used to create CLS, PCR, and PLS models. MCR-ALS Toolbox Version 2.1 (Universitat de Barcelona, Barcelona, Spain) was used for MCR-ALS analysis. GA-PLS and FA-PLS models were implemented using the Global Optimization Toolbox in MATLAB R2023a. Custom MATLAB scripts utilizing the Statistics Toolbox (MathWorks Inc., Natick, MA, USA) constructed Brereton’s multilevel calibration design and MMD validation set. Student’s t-test, one-way ANOVA, and Elliptical Joint Confidence Region (EJCR) analysis were performed using proprietary MATLAB algorithms.

Materials and chemicals

Standards

PIR and VEN were generously supplied by the National Organization for Drug Control and Research (NODCAR, Cairo, Egypt). The purity of PIR was confirmed according to the official United States Pharmacopeia monograph for PIR [26], while VEN purity was established using the validated RP-HPLC procedure described by Somasekhar et al. [27]. The certified purity values were 99.21% ± 0.5 for PIR and 99.41% ± 0.3 for VEN. Prior to analysis, both reference materials were maintained in tightly closed containers under the storage conditions specified by the respective manufacturers.

Pharmaceutical formulations

Rytreka^® tablets (batch No. 816A1) was obtained from Serve Pharmaceuticals, India and is labeled to contain 25 mg PIR and 75 mg VEN per tablet.

Chemicals and solvent system

All aqueous solutions were prepared with ultra-pure water generated by a Milli-Q IQ 7000 purification system (Millipore, Burlington, MA, USA). The system produced water with a resistivity of 18.2 MΩ·cm at 25 °C and total organic carbon below 2 ppb. The purified water was passed through 0.22 μm membrane filters before use in solution preparation to minimize particulate contamination during spectrophotometric measurements. Absolute ethanol (≥ 99.8%) and HPLC-grade methanol were purchased from Sigma-Aldrich (St. Louis, MO, USA). Unless otherwise specified, the solvent used for preparing the working solutions was a 1:1 (v/v) mixture of ultra-pure water and ethanol. Methanol was restricted to the plasma pretreatment step, where it was used for protein precipitation.

Biological samples

Blank human plasma was obtained from the Egyptian Holding Company for Biological Products and Vaccines (VACSERA, Giza, Egypt). Samples were obtained from healthy participants in compliance with institutional ethical standards and preserved at − 80 °C until utilized for further investigation.

Preparation of standard solutions

Separate PIR and VEN stock solutions were made at 1.00 mg mL⁻¹ by accurately transferring 50.0 mg of each reference standard into 50.0 mL volumetric flasks and dissolving in a 1:1 water-ethanol combination. At room temperature, ultrasonic agitation for 20 min disintegrated everything. Stock solutions were cooled (2–8 °C) in amber glass containers until needed.

Through serial dilution with the same water–ethanol (1:1, v/v) solvent system, functional solutions were created. Initially, 100.0 µg mL⁻¹ intermediate solutions were created to reduce cumulative volumetric error during dilution steps. Calibration and validation mixes covering the study concentration ranges were made from the solutions. The volumetric measurements were taken with Class A calibrated glassware.

Spectral range selection and concentration domain

PIR and VEN were first examined individually by recording their zero-order UV absorption spectra from 200 to 450 nm against a water–ethanol mixture (1:1, v/v) as the blank (Fig. 1). Measurements were obtained from standard solutions containing 5.0 µg mL⁻¹ PIR and 25.0 µg mL⁻¹ VEN. The two spectra exhibited extensive overlap over a large portion of the recorded region, making direct spectrophotometric resolution of the two compounds impractical and supporting the use of multivariate calibration.

Fig. 1.

Zero-order absorption spectra of VEN and PIR over the investigated wavelength range

For CLS, PCR, PLS, GA-PLS, and FA-PLS, the wavelength variables were selected from 210 to 400 nm, resulting in 191 variables for model construction. MCR-ALS was treated differently: the entire 200–450 nm range was retained to avoid discarding spectral information relevant to the bilinear decomposition and estimation of the individual component profiles.

The concentration domain used for multivariate modeling covered 5.0–30.0 µg mL⁻¹ for VEN and 1.0–9.0 µg mL⁻¹ for PIR. Calibration and validation mixtures were prepared within these respective ranges. Consequently, all spectra entering model development and subsequent validation were generated within the defined concentration domain and, for the regression-based models, within the selected 210–400 nm spectral interval.

Experimental design and validation set generation

Two independent experimental domains were established: one for pharmaceutical formulation analysis and another for matrix-matched plasma analysis. Each domain comprised a dedicated calibration set and validation set prepared separately and processed independently during chemometric model development.

Calibration samples were generated using Brereton’s multilevel design, with the aim of distributing the mixtures systematically over the concentration domain relevant to the analysis. For each matrix, 25 binary mixtures were prepared by assigning five concentration levels (− 2, − 1, 0, + 1, +2) independently to VEN and PIR. The resulting concentration domains were 5.0–30.0 µg mL⁻¹ for VEN and 1.0–9.0 µg mL⁻¹ for PIR, corresponding to the validated working ranges of the spectrophotometric procedure. Thus, the calibration set covered the specified concentration space in a balanced manner rather than concentrating samples at only a few levels.

Independent validation mixtures were produced via MMD as a space-filling design strategy to provide spatially varied compositions within the identical concentration range, while ensuring independence from calibration coordinates. Candidate mixture formulations were initially produced by continuous sampling throughout the specified experimental domain, subsequently eliminating any coordinates related to calibration mixes.

The MMD approach was implemented as a geometric space-filling criterion, wherein validation compositions were chosen to maximize the minimal Euclidean distance across all validation points:

1

where and signify mixture compositions inside the experimental domain , and denotes the Euclidean distance metric. This criterion encourages consistent spatial distribution of validation samples across the concentration spectrum.

The methodical selection of validation combinations adhered to the iterative principle:

2

where denotes the incompletely assembled validation set at iteration t. Iterations persisted until the specified quantity of validation mixtures was achieved. In each analytical domain, the ultimate validation set comprised 16 mixes, allocated throughout the experimental area without repeating calibration coordinates. Calibration and validation mixtures for the pharmaceutical sector were formulated through the suitable dilution of standard solutions utilizing a water–ethanol (1:1, v/v) solvent system. In the plasma sector, matrix-matched combinations were developed with unadulterated human plasma specimens that underwent protein precipitation before the introduction of analytes.

Chemometric model development

Six multivariate calibration approaches representing classical regression, latent-variable modeling, metaheuristic optimization for variable selection, and curve-resolution methodologies were implemented for the quantitative analysis of the (PIR–VEN) system. The evaluated models comprised CLS, PCR, PLS, GA-PLS, FA-PLS, and MCR-ALS. Before model development, spectral datasets obtained within the selected wavelength interval were subjected to baseline correction and mean-centering. Autoscaling procedures were applied when required to normalize variable magnitudes and stabilize multivariate computations. Model calibration was performed using the designed calibration mixtures, while predictive evaluation employed the independently generated validation sets for both pharmaceutical and plasma domains.

Classical least squares

Beer–Lambert linear additive was used to create CLS calibration models. This method expressed the spectral response matrix as a linear mixture of pure component spectra weighted by analyte concentrations. Least-squares minimization of residual error between measured and reconstructed spectra calculated model parameters.

Principal component regression

PCR modeling was performed through dimensionality reduction of the spectral matrix using principal component analysis (PCA). Orthogonal latent variables (LVs) were generated via singular value decomposition of the mean-centered data matrix. Regression of analyte concentration vectors was subsequently carried out against the selected principal component scores. The number of retained LVs was determined through cross-validation.

Partial least squares

PLS regression was applied to establish predictive relationships between spectral variables and analyte concentrations. The PLS algorithm simultaneously decomposed the spectral matrix and concentration matrix to generate LVs maximizing covariance between predictor and response matrices. Model dimensionality was optimized through cross-validation to get the appropriate number of LVs.

Genetic algorithm–partial least squares

GA-PLS selected wavelengths before PLS modeling. Candidate subsets of spectral variables were encoded as chromosomes within a genetic population. Evolutionary operations including selection, crossover, and mutation were applied iteratively to identify wavelength subsets minimizing cross-validated prediction error. The optimized variable subset was subsequently used as the input set for the final PLS regression model.

PLS modeling with firefly-based wavelength selection

The FA-PLS method identified quantitative prediction-effective spectral areas. Each firefly was treated as a possible combination of wavelengths extracted from the full spectral matrix, and the quality of a given combination was judged from the corresponding cross-validated prediction error. The firefly population was subsequently updated according to the relative attractiveness of the candidate solutions and their model fitness. This search was repeated iteratively until the predefined convergence condition was reached. The wavelengths retained at convergence were finally used as the input variables for construction of the PLS calibration model.

Resolution of overlapped spectra by MCR-ALS

Following a bilinear model, MCR-ALS separated the measured spectral matrix into its concentration and spectral components. Initial concentration profile estimates were made using Evolving Factor Analysis. These estimations were revised using alternating least-squares optimization, with concentration and spectral profile non-negativity applied to maintain chemically relevant solutions. The alternating optimization was repeated until the residual matrix stopped changing, indicating solution stabilization.

Selection criteria of models

Interrogating the PIR–VEN system through complementing algorithmic viewpoints like linear regression, metaheuristic optimization, and bilinear resolution led multivariate calibration framework selection.

CLS was included as a direct linear calibration model based on the Beer–Lambert additive relationship between component spectra and mixture responses. Latent-variable regression methods (PCR and PLS) were implemented to address spectral collinearity and dimensionality reduction within the multivariate dataset. Metaheuristic optimization approaches (GA-PLS and FA-PLS) were employed for wavelength-variable selection prior to PLS calibration. The MCR-ALS framework was incorporated as a bilinear resolution method capable of decomposing the spectral matrix into concentration and pure spectral profiles. This multi-model strategy enabled evaluation of the analytical system using regression-based, metaheuristic variable-selection, and curve-resolution methodologies within the same experimental dataset.

Validation of the chemometric models and analytical performance

The validation scheme was designed around the requirements of ICH Q2(R1) for analytical procedure validation [28], while also considering the principles introduced in the revised ICH Q2(R2) and Q14 guidelines concerning modern analytical procedures, analytical development, and lifecycle-based method management [29]. Particular attention was given to the performance of the multivariate calibration models rather than relying solely on conventional univariate validation parameters. Calibration and independent validation samples generated from the experimental design were therefore used to examine model accuracy, precision, sensitivity, robustness, and selectivity [30]. Predictive errors were characterized by comparing the reference concentration vector (y i ) with the corresponding predicted vector (ŷ i ) for n samples. External validation was performed using 16 independently prepared mixtures for each analytical matrix. RMSEC, RMSEP, RRMSEP, bias, BCRMSEP, and SEC were calculated as complementary measures of model performance. These parameters collectively describe the deviation of the calculated concentrations from their reference values, the predictive error associated with new samples, and the extent of systematic error in the calibration and prediction results.

The mathematical expressions applied to calculate these statistical parameters are given below:

 

where represents the mean reference concentration.

Sensitivity at low analyte levels was investigated through LOD and LOQ calculations using the NAS approach. This procedure separates the spectral contribution attributable to the analyte from the remaining multicomponent signal, making it particularly suitable for quantitative assessment when the spectra of the investigated compounds substantially overlap.

Accuracy was examined by recovery experiments conducted at three levels distributed across the validated concentration range. The validation mixtures contained VEN at 10.0, 17.5, and 25.0 µg mL⁻¹ and PIR at 2.0, 5.0, and 8.0 µg mL⁻¹. Six independent measurements were obtained at each concentration level. The measured concentrations were subsequently expressed as percentages of the corresponding nominal values to calculate recovery.

Precision was investigated under two experimental conditions. Repeatability was determined from six replicate measurements carried out during the same analytical session while maintaining the operating conditions unchanged. Intermediate precision was examined by repeating the entire procedure on three successive days, with independent solutions freshly prepared for each day. This design allowed the contribution of normal day-to-day variation to the analytical results to be evaluated.

Robustness was tested by deliberately altering selected spectrophotometric acquisition conditions within narrow limits. The investigated factors were spectral bandwidth (1.0 ± 0.2 nm), scan velocity (slow, medium, and fast scanning modes), and repeated removal and reinsertion of the quartz cuvette, the latter being used to examine the influence of cuvette positioning and alignment. Predicted concentrations of PIR and VEN were compared under the modified conditions. The procedure was regarded as robust when recoveries remained within 98.0–102.0% and %RSD was ≤ 2.0%.

Selectivity was evaluated in the presence of the components expected to accompany PIR and VEN in the analyzed matrices. Accordingly, the possible spectral effects of pharmaceutical excipients and plasma constituents were examined. The resulting spectra were evaluated with the developed chemometric models to confirm that the contributions of PIR and VEN could be resolved quantitatively despite the presence of potentially interfering matrix signals.

Elliptical joint confidence region (EJCR) analysis

The six chemometric models were compared statistically using the EJCR procedure [31]. This method treats the regression slope (b) and intercept (a) as a pair rather than evaluating them separately, allowing systematic deviations from the ideal relationship to be examined through their joint uncertainty. Consequently, both proportional and constant bias can be assessed within a single statistical framework.

For each analyte, concentrations predicted for the validation mixtures by each chemometric model were regressed against the corresponding reference concentrations. The resulting estimates of a and b were then used to construct the 95% confidence region from their variance–covariance matrix. The resulting ellipse describes the joint uncertainty associated with the two regression coefficients, with its position and extent providing a basis for evaluating agreement between predicted and reference concentrations.

The EJCR was calculated using the following quadratic form:

Assuming 𝜃=(𝑎,𝑏) for the intercept-slope parameter vector,, 𝜃. for the estimated regression coefficients, Σ for the variance-covariance matrix, and,𝜒-2, 0.95-2.The 95% confidence level chi-square critical value with two degrees of freedom. The estimated confidence ellipses were shown in intercept–slope coordinate space relative to the theoretical ideal point (b = 1, a = 0) reflecting perfect predicted-reference concentration agreement. Eigenvalue decomposition of the covariance matrix, which describes the regression parameters’ primary directions of uncertainty, yielded ellipse geometry, orientation, and axis magnitude. EJCR visualization and numerical computation using bespoke MATLAB statistical methods. The created calibration models were assessed for proportional deviation and constant systematic bias using this multivariate inferential technique in addition to conventional error measurements.

Assay of VEN and PIR in the combined tablet formulation

The method quantified VEN and PIR simultaneously in the marketed combination tablet dose form. A 100 mL volumetric flask contained accurate-weighed finely powdered tablets containing 75.0/25.0 mg VEN and PIR. The powder was ultrasonically extracted with water–ethanol (1:1, v/v) for 30 min. After cooling to ambient temperature, the extract was diluted with the same solvent and filtered through a 0.45 μm membrane.

To achieve values within the validated ranges of 5.0–30.0 µg mL⁻¹ for VEN and 1.0–9.0 µg mL⁻¹ for PIR, a portion of the filtrate was diluted with the extraction solvent. Under optimal conditions, spectra were acquired across 210–400 nm, and chemometric models quantified the two analytes simultaneously. Standards of 80%, 100%, and 120% of nominal concentration were also added to verify accuracy. Six replicates were taken at each fortification. %RSD, bias, and recovery were reported.

Application to spiked human plasma

Preparation of sets

Drug-free human plasma samples rich in properly measured PIR and VEN were analyzed in a biological matrix to test the approach. To maintain homogeneity, frozen plasma was gently homogenized at room temperature before sample processing. Blank plasma spectra were first obtained over the desired wavelength range to describe the endogenous spectral background and build matrix-specific calibration models.

Plasma spectral contribution was accounted for by matrix-specific calibration. According to the experimental design, 25 fortified plasma mixtures were calibrated and 16 separately generated samples were validated. The concentration combinations covered plasma PIR and VEN validation ranges.

Plasma sample pretreatment

Plasma preparation by protein precipitation reduced endogenous protein input to spectrophotometric measurements. In centrifuge tubes, 1.0 mL plasma aliquots were combined with 3.0 mL methanol as the protein-precipitating solvent. The mixtures were vortexed for 60 s and centrifuged at 4000 × g for 10 min at 4 °C.

Centrifugation separated the clear supernatants, which were dried under a moderate nitrogen stream at 40 °C. After rehydrating each residue in 1.0 mL of water–ethanol (1:1, v/v), spectrophotometric measurements were taken. Reconstituted materials were filtered using 0.22 μm syringe filters before spectrum capture.

Determination of analytes in fortified human plasma

After completion of the plasma pretreatment procedure, the resulting extracts were diluted, where required, so that the concentrations of the analytes fell within the established calibration ranges for the plasma matrix. The prepared solutions were then subjected to spectrophotometric measurement using the instrumental conditions described previously. Quantification was performed with the chemometric calibration models developed specifically for the plasma samples.

Analytical performance was assessed for all investigated chemometric models using independently prepared fortified plasma samples. Additionally, the optimized MCR-ALS model was employed to conduct a preliminary assessment of matrix influence by comparing post-extraction spiked plasma samples with neat standard solutions prepared at identical concentrations. Matrix factors and matrix effects were determined to assess the remaining contribution of endogenous plasma constituents following the optimized protein precipitation protocol. This experiment was intended to provide a preliminary indication of matrix influence under the investigated conditions and was not designed as a comprehensive regulatory bioanalytical matrix-effect assessment. It is important to emphasize that the plasma investigation was performed exclusively using laboratory-prepared spiked human plasma samples. Accordingly, the plasma experiments were considered a proof-of-concept evaluation of the developed spectrophotometric–chemometric procedure in a biological matrix. They were not intended to constitute full bioanalytical validation for routine clinical use or pharmacokinetic studies.

Results and discussion

Establishment of a greener solvent system

Solvent screening using the green solvent selection tool (GSST)

Solvent selection was incorporated into the method-development process from the outset, with environmental and practical considerations evaluated alongside the expected analytical behavior. The candidate solvents were screened using the GSST model reported by Larsen et al. [32]. Seven solvents with diverse polarities were compared. In Fig. S2, water had the greatest greenness score (G = 7.3), followed by ethyl acetate (G = 6.7) and ethanol (G = 6.6). Methanol and acetonitrile scored the same (G = 5.8), whereas hexane and chloroform scored lower due to their poor environmental, health, and safety profiles. GSST results narrowed solvent selection to better choices before experimental optimization. A water–ethanol mixture (1:1, v/v) was chosen as the solvent system for the suggested process.

Additional solvent evaluation by spider diagram analysis

The solvent candidates were further examined using SDAGI to compare their sustainability profiles according to SHE criteria obtained from Safety Data Sheets [33, 34]. The corresponding results are summarized in Table S1 and Fig. 2. Ethanol showed the most favorable overall profile with respect to environmental compatibility and handling safety. Methanol and acetonitrile were less attractive because of their comparatively greater toxicity and associated handling concerns. Ethyl acetate also displayed acceptable sustainability characteristics; however, its analytical behavior was less favorable under the investigated spectrophotometric conditions.

Fig. 2.

SDAGI spider diagram comparing the greenness profiles of the investigated solvents

Direct spectral evaluation showed that ethanol produced stronger absorbance responses and better separation of the spectral features of both analytes. This analytical advantage, together with its favorable sustainability profile, led to the selection of ethanol for the subsequent method-development experiments. Thus, GSST and SDAGI were used as complementary screening tools, with the final solvent choice based on both solvent sustainability and its suitability for the intended spectrophotometric measurements.

Chemometric model development and spectral dataset design

Spectral characteristics of the VEN–PIR system

Both substances have strong UV absorption and spectrum overlaps in the analytical range, especially between 210 and 300 nm (Fig. 1), making direct spectrophotometric determination problematic. Traditional univariate methods, such as λmax measurements, ratio-spectrum processing, derivative spectrophotometry, and dual-wavelength measurements, led to insufficient selectivity due to spectral overlap and insufficient wavelength regions. Spectra were taken from 200 to 450 nm to assess each analytes’ full spectrum characteristics. Baseline stability, solvent interference, signal-to-noise ratio, and spectral information content showed that the regression models’ strongest spectral variables were in the 210–400 nm interval. The CLS, PCR, PLS, GA-PLS, and FA-PLS models were built from the selected spectral region. Spectral data were taken at 1 nm intervals, yielding 191 wavelength variables per sample. The region below 210 nm was eliminated to avoid solvent background absorption and experimental noise, and wavelengths beyond 400 nm were ignored because they gave little quantitative information.

In contrast, MCR-ALS analysis used the entire 200–450 nm spectral range. MCR-ALS preserves the full spectral information to recover pure spectral profiles and concentration distributions while minimizing rotational ambiguity by bilinear decomposing the entire spectral matrix rather than regression on selected variables. The regression-based models’ optimized wavelength selection and MCR-ALS’s complete spectral domain provided a robust analytical framework for resolving the highly overlapping VEN–PIR spectral system while maximizing each chemometric approach’s strengths.

Matrix-specific chemometric modeling

Due to their different spectrum properties, each analytical matrices have separate chemometric models. Pharmaceutical excipients cause little spectral interference, while endogenous substances in human plasma add background absorption and matrix-dependent spectral variability. Thus, different calibration models were created for each matrix to optimize quantitative performance under their analytical conditions.

Brereton’s multilevel experimental design was used to generate 25 calibration mixtures for each matrix with concentration combinations throughout VEN and PIR ranges. Then, 16 MMD-selected validation combinations were used to evaluate predictive performance. This selection represented the experimental design space without using calibration mixtures. Different calibration and validation sets were created for the two matrices to account for their spectrum differences and eliminate matrix-related bias in model evaluation.

Using matrix-specific datasets, the six chemometric models (CLS, PCR, PLS, GA-PLS, FA-PLS, and MCR-ALS) were built and tested. Quantitative predictions in pharmaceutical formulations and spiked human plasma were more accurate and resilient because this modeling technique adjusted calibration settings and variable-selection procedures to matrix spectral features.

Calibration dataset design using brereton’s multilevel strategy

The calibration dataset was established using Brereton’s multilevel experimental design, which generates statistically balanced sample distributions suitable for multivariate spectroscopic calibration. For uniform representation throughout verified analytical working ranges, 25 binary mixtures were created by mixing five coded concentration levels (− 2, − 1, 0, + 1, and + 2) individually assigned to VEN and PIR.

The resulting calibration matrix incorporated central, intermediate, and extreme concentration combinations, ensuring adequate representation across the experimental design space while decreasing correlation between the concentrations of the two analytes. This balanced distribution enhances multivariate model development by ensuring that spectral variation is predominantly associated with compositional changes rather than uneven sampling of the concentration domain.

The complete calibration design is presented in (Table 1).

Table 1.

Brereton multilevel calibration design (n = 25) and MMD external validation design (n = 16) used for chemometric model development

Mix number	Calibration set (µg mL⁻¹)
	VEN	PIR
1.	15	5
2.	15	1
3.	5	1
4.	5	9
5.	30	3
6.	10	9
7.	30	5
8.	15	3
9.	10	3
10.	10	7
11.	25	9
12.	30	7
13.	25	5
14.	15	9
15.	30	9
16.	30	1
17.	5	7
18.	25	1
19.	5	5
20.	15	7
21.	25	7
22.	25	3
23.	10	1
24.	5	3
25.	10	5

Mix number	Validation set (µg mL⁻¹)
	VEN	PIR
1.	20	3
2.	22	8
3.	8	7
4.	22	9
5.	16	4
6.	6	2
7.	11	7
8.	16	8
9.	26	5
10.	28	6
11.	10	6
12.	29	3
13.	25	4
14.	14	2
15.	18	5
16.	9	1

External validation dataset generated by MMD

MMD was used as a space-filling design strategy that selects compositions so that the closest pair remains as far apart as possible, thus guaranteeing representative and well-distributed coverage of the experimental concentration space.

The validation design adopted in this study is summarized schematically in (Fig. 3). After constructing the calibration dataset, all calibration compositions were removed from the candidate pool to preserve complete independence between calibration and validation samples. The MMD-based selection procedure was subsequently employed to select 16 validation mixtures distributed throughout the concentration domain defined by VEN and PIR.

Fig. 3.

Workflow of the Maximin Distance Design (MMD) employed for algorithmic optimization of the external validation set

The spatial distributions of the calibration and validation datasets are illustrated in (Figs. 4 and 5). These validation samples included both center and boundary parts of the experimental design space, allowing a representative assessment of model predictive ability across concentration ranges. Moreover, the comparable concentration distributions observed for the calibration and validation datasets indicate the absence of systematic sampling bias.

Fig. 4.

Two-dimensional distribution of the MMD validation mixtures illustrating uniform space-filling coverage of the analytical domain

Fig. 5.

Radar and boxplot distribution profiles of calibration and validation mixtures for VEN and PIR, demonstrating comparable concentration coverage and uniform spatial distribution within the analytical domain

The combined implementation of Brereton’s multilevel calibration design and MMD-based validation design generated independent calibration and validation datasets, thus, the prediction performance of the explored chemometric models could be assessed consistently and statistically.

Development and comparative evaluation of multivariate calibration models

Classical least squares

CLS was first established as the reference multivariate calibration model for the simultaneous quantification of VEN and PIR. Model construction was based on 25 Brereton multilevel calibration mixtures and the optimized spectral region spanning 210–400 nm (191 wavelength variables). Before calibration, the spectral data were subjected to mean-centering and autoscaling to improve numerical stability and model performance.

Since CLS directly relies on the Beer–Lambert relationship without employing latent-variable extraction or variable-selection procedures, all recorded wavelength variables were incorporated into the calibration model. Consequently, its predictive capability is strongly influenced by spectral linearity and is comparatively more sensitive to spectral collinearity in systems exhibiting extensive overlap. Nevertheless, CLS provides an appropriate benchmark against which latent-variable-based and optimization-assisted chemometric models can be objectively compared.

Principal component regression

To minimize the influence of spectral collinearity, PCR was developed using the mean-centered and autoscaled spectral matrix. The number of latent variables was selected through leave-one-out cross-validation, using the minimum RMSECV as the criterion for model optimization (Fig. S3A). The resulting RMSECV profile showed that prediction errors reached a stable minimum after the inclusion of three latent variables. Consequently, three latent variables were chosen, providing efficient representation of the relevant spectral information while avoiding unnecessary model complexity.

Partial least squares

Using the preprocessed spectral matrix, the PLS calibration model was constructed, and its latent-variable dimensionality was decided by leave-one-out cross-validation from the resulting RMSECV profile (Fig. S3B). As observed for PCR, prediction errors stabilized after three latent variables, and this model complexity was therefore adopted for subsequent analyses. Unlike PCR, PLS simultaneously incorporates spectral and concentration information during latent-variable extraction, producing calibration models that are directly optimized for quantitative prediction.

Genetic algorithm–PLS

To enhance model selectivity and remove redundant spectral information, GA-based variable selection was integrated with PLS regression. In order to pick the wavelength regions from the whole spectral matrix that contributed the most efficiently to the prediction, GA was utilized prior to the construction of the models. Table S2 contains a summary on the GA parameters that have been optimized.

For the purpose of selecting the model complexity, leave-one-out cross-validation was utilized, and the RMSECV was utilized as the criterion via which optimization was performed. As shown in (Fig. S3C), the prediction error stabilized after two latent variables; therefore, two LVs were chosen. Relative to the full-spectrum PCR and PLS models, GA-PLS achieved comparable predictive performance while requiring fewer latent variables.

The optimized GA-PLS models also produced substantial spectral compression. Of the original 191 wavelength variables, only 79 were retained for VEN and 96 for PIR, corresponding to dimensionality reductions of 58.6% and 49.7%, respectively. The larger reduction observed for VEN is consistent with its more localized absorption characteristics, whereas the broader spectral profile of PIR required retention of a greater number of informative wavelengths.

Overall, incorporating GA into the variable-selection process effectively reduced spectral dimensionality.

Firefly algorithm–optimized PLS

Additional wavelength optimization was performed by coupling FA with PLS regression. The optimized FA parameters are presented in (Table S3).

The number of latent variables was selected through leave-one-out cross-validation. In Fig. S3D, the RMSECV curve shows little further improvement beyond the second latent variable; therefore, two latent variables were selected for VEN and PIR.

The optimized FA-PLS models also provided stronger variable reduction than GA-PLS. Of the original 191 wavelength variables, FA selected 63 for VEN and 74 for PIR, corresponding to reductions of 67.0% and 61.3%, respectively. Despite this substantial reduction in dimensionality, the selected wavelength subsets retained the spectral information needed for quantitative prediction. Relative to the full-spectrum PLS model, FA-PLS reduced both model complexity and computational demand by eliminating non-informative wavelength variables without compromising predictive performance. Consequently, the optimized wavelength subsets provided a simplified yet information-rich spectral representation suitable for subsequent quantitative analysis.

Resolving spectral contributions by MCR-ALS

MCR-ALS was applied to the experimental spectral matrix to separate the contributions of VEN and PIR despite their pronounced spectral overlap. The procedure represents the measured data through a bilinear model, from which concentration profiles and the corresponding pure spectral profiles are estimated. Model development was carried out using the 25 calibration mixtures produced in accordance with Brereton’s multilevel framework within the complete recorded spectral range (200–450 nm).

Before ALS optimization, the appropriate number of components was estimated by Evolving Factor Analysis, which recognized two major elements associated with the two analytes. The following ALS optimization was conducted by enforcing non-negativity restrictions on the concentration matrices as well as the spectrum matrices, in addition to closure restrictions for the concentration profiles, in order to ensure that the solutions are chemically valid. Repetitive computations persisted until the residual variance achieved convergence.

The resolved spectral profiles are presented in (Fig. 6) together with the corresponding experimentally recorded reference spectra, whereas the plasma results are shown in (Fig. S4). Excellent agreement was observed between the resolved and experimental spectra for both analytes, demonstrating successful recovery of their characteristic spectral features despite the severe overlap in the original absorption spectra. Comparable spectral reconstruction was also achieved for spiked human plasma samples following protein precipitation, confirming the ability of MCR-ALS to separate analyte signals from matrix-associated background contributions.

Fig. 6.

Experimental and MCR-ALS-resolved spectra of VEN and PIR showing spectral resolution of the overlapped binary system

Overall, MCR-ALS provided an effective alternative to conventional regression-based calibration by directly resolving overlapping spectral information into chemically meaningful concentration and spectral profiles. The resolved spectra were subsequently utilized for quantitative analysis alongside the other investigated chemometric models.

Model validation and predictive performance

Independent external validation

An independent set of 16 mixtures was used to examine the prediction performance of the developed chemometric models. The mixtures were generated using the MMD algorithm and covered the complete working ranges of VEN (5.0–30.0 µg mL⁻¹) and PIR (1.0–9.0 µg mL⁻¹), with compositions distributed across the center, intermediate regions, and boundaries of the calibration domain (Table 1). Such distribution allowed model predictions to be examined over a broad range of compositions rather than at a limited portion of the experimental space.

The selection procedure was based on geometric separation between candidate mixtures. Specifically, MMD favors compositions that maintain large minimum Euclidean distances from one another, thereby reducing the formation of closely grouped validation points and promoting more even representation of the concentration domain. This design provides a more demanding basis for evaluating the robustness and external predictive behavior of the chemometric models. Consequently, model performance was evaluated using mixtures representative of the entire analytical space rather than a limited portion of the calibration domain. This design enhances the reliability of external validation by providing a realistic measure of interpolation performance across the full experimental range.

Complete independence between the calibration and validation datasets further minimized the possibility of overly optimistic prediction statistics resulting from compositional redundancy. Accordingly, Brereton’s multilevel calibration design and MMD-based external validation created a statistically balanced and information-rich experimental environment for unbiased chemometric model comparison.

Comparative performance

Table 2 summarizes the analytical performance of the chemometric models for calibration and validation datasets. A clear enhancement in predictive capability was observed with the progression from conventional full-spectrum regression approaches to latent-variable methods, optimization-guided wavelength selection strategies, and ultimately bilinear spectral resolution.

Table 2.

Calibration and validation performance of the investigated chemometric models for the determination of VEN and PIR in pharmaceutical formulations and human plasma

For analytes Determination in dosage form			CLS		PCR		PLS		GA-PLS		FA-PLS		MCR-ALS
			VEN	PIR	VEN	PIR	VEN	PIR	VEN	PIR	VEN	PIR	VEN	PIR
	calibration set	MEAN	98.74	98.31	99.16	98.94	99.42	99.21	99.63	99.48	99.81	99.67	99.94	99.88
		SD	0.62	0.78	0.51	0.66	0.44	0.57	0.36	0.45	0.28	0.36	0.21	0.29
		%RSD	0.63	0.79	0.52	0.67	0.44	0.58	0.36	0.45	0.28	0.37	0.21	0.29
		RMSEC(a)	0.684	0.173	0.541	0.142	0.437	0.118	0.361	0.094	0.284	0.073	0.211	0.054
	validation set	MEAN	98.52	98.08	99.01	98.74	99.31	99.06	99.54	99.37	99.76	99.61	99.91	99.84
		SD	0.74	0.86	0.62	0.75	0.52	0.63	0.42	0.52	0.34	0.42	0.26	0.34
		%RSD	0.75	0.88	0.63	0.76	0.52	0.64	0.43	0.52	0.34	0.42	0.26	0.34
		RMSEP(b)	0.792	0.201	0.628	0.168	0.514	0.139	0.428	0.116	0.337	0.091	0.259	0.071
For analytes Determination in Plasma			CLS		PCR		PLS		GA-PLS		FA-PLS		MCR-ALS
			VEN	PIR	VEN	PIR	VEN	PIR	VEN	PIR	VEN	PIR	VEN	PIR
	calibration set	MEAN	96.42	96.11	97.38	97.04	98.24	98.02	98.86	98.62	99.24	99.11	99.71	99.58
		SD	1.84	1.96	1.52	1.68	1.22	1.38	1.05	1.19	0.82	0.96	0.64	0.78
		%RSD	1.91	2.04	1.57	1.73	1.24	1.41	1.06	1.20	0.83	0.97	0.64	0.79
		RMSEC(a)	1.384	0.418	1.127	0.347	0.894	0.281	0.714	0.224	0.541	0.168	0.382	0.116
	validation set	MEAN	96.18	95.92	97.04	96.86	97.96	97.74	98.61	98.42	99.08	98.94	99.62	99.48
		SD	2.08	2.24	1.74	1.92	1.38	1.56	1.18	1.34	0.94	1.08	0.74	0.86
		%RSD	2.17	2.34	1.80	1.99	1.41	1.60	1.20	1.36	0.95	1.10	0.75	0.87
		RMSEP(b)	1.621	0.506	1.318	0.418	1.041	0.336	0.842	0.271	0.648	0.204	0.468	0.141

^aRoot Mean Square Error of Calibration

^bRoot Mean Square Error of prediction

The CLS model exhibited the weakest predictive performance, particularly for plasma samples, highlighting its sensitivity to spectral collinearity and interference arising from the complex biological matrix. Incorporation of latent-variable techniques through PCR and PLS improved analytical performance by compressing redundant spectral information into a reduced set of relevant variables. Additional gains were achieved with the optimization-assisted models, GA-PLS and FA-PLS, where the selection of informative wavelength regions minimized prediction errors and enhanced recovery values.

Among all evaluated approaches, MCR-ALS showed the best overall numerical performance, as reflected by the lowest RMSEC and RMSEP values and the most favorable recovery and precision results for both pharmaceutical formulations and plasma samples (Table 2). Its high performance can be attributed to the effective resolution of overlapping spectral profiles before quantification, thereby minimizing spectral interference and enhancing prediction accuracy.

Performance differences among the investigated models became more evident in spiked human plasma than in pharmaceutical formulations, reflecting the greater spectral complexity associated with the biological matrix. Despite this increased analytical challenge, all chemometric approaches produced acceptable results, whereas the optimization-assisted models and, particularly, MCR-ALS demonstrated the highest predictive reliability under the investigated experimental conditions.

Comprehensive validation of the proposed analytical method

Method sensitivity was tested using NAS, as shown in Tables S4 and S5. Pharmaceutical formulations showed lower LOD values for VEN (0.413 µg mL⁻¹ with CLS to 0.125 µg mL⁻¹ with MCR-ALS) and lower LOD values for PIR (0.098 to 0.029 µg mL⁻¹). The LOQ values for VEN ranged from 1.251 to 0.378 µg mL⁻¹, while PIR ranged from 0.298 to 0.087 µg. The tested models’ analytical sensitivity improved when LOD and LOQ decreased from CLS to MCR-ALS. Endogenous spectrum interference and matrix complexity increased plasma detection limits as expected. In this matrix, VEN LOD values ranged from 0.862 to 0.245 µg mL⁻¹, while PIR values ranged from 0.215 to 0.058 µg. Even with this increase, the sensitivity was suitable for the plasma concentration range.

Calibration quality and predictive performance were further evaluated using RMSEC, RMSEP, RRMSEP, BCRMSEP, and SEC. A consistent reduction in RMSEC was observed when progressing from CLS through PCR, PLS, GA-PLS, and FA-PLS to MCR-ALS, indicating progressively improved calibration performance. For pharmaceutical formulations, RMSEC values for VEN decreased from 0.684 to 0.211, while those for PIR declined from 0.173 to 0.054. Similar improvements were observed for plasma samples, where RMSEC values decreased from 1.384 to 0.382 for VEN and from 0.418 to 0.116 for PIR. External prediction followed the same trend. In pharmaceutical formulations, RMSEP values decreased from 0.792 to 0.259 µg mL⁻¹ for VEN and from 0.201 to 0.071 µg mL⁻¹ for PIR, whereas in plasma they decreased from 1.621 to 0.468 µg mL⁻¹ for VEN and from 0.506 to 0.141 µg mL⁻¹ for PIR. These results indicate improved predictive performance across the investigated chemometric models.

The RRMSEP values also reflected the improvement in proportional prediction accuracy. In pharmaceutical formulations, RRMSEP decreased from 2.85% to 0.93% for VEN and from 3.17% to 1.14% for PIR. For plasma, RRMSEP decreased from 5.81% to 1.68% for VEN and from 6.47% to 1.96% for PIR. Thus, although the plasma matrix produced higher proportional prediction errors, the same improvement across the chemometric models was observed. In all cases, BCRMSEP values less than RMSEP values, indicating a limited contribution of bias to the overall prediction error. Similarly, SEC values decreased progressively toward MCR-ALS, supporting improved calibration performance.

Accuracy was assessed from the recovery results obtained using the external validation measurements. In pharmaceutical formulations, mean recoveries ranged from 99.48 to 99.85% for VEN and 99.45–99.87% for PIR, with SD values of 0.70–1.02% and 0.67–1.08%, respectively. For spiked plasma, mean recoveries ranged from 99.52 to 99.84% for VEN and 99.47–99.86% for PIR, with SD values of 0.68–1.08% and 0.65–1.04%, respectively. All mean recoveries were therefore within the specified 98–102% range, while the corresponding SD values remained below 2%. Among the investigated models, MCR-ALS recovered closest to 100% in both matrices, although the differences among the models were relatively small.

Precision was examined at two levels, namely repeatability and intermediate precision. In pharmaceutical formulations, repeatability ranged from 0.19 to 0.59% RSD for VEN and 0.28–0.74% RSD for PIR, whereas intermediate precision ranged from 0.24 to 0.67% RSD and 0.33–0.83% RSD, respectively. For plasma samples, repeatability ranged from 0.68 to 1.76% RSD for VEN and 0.81–1.95% RSD for PIR, while intermediate precision ranged from 0.76 to 1.93% RSD and 0.89–2.10% RSD, respectively. Plasma’s increased variability reflects the biological matrix’s complexity. Robustness results showed RSD values of 0.47–2.32% for the investigated plasma measurements and 0.32–0.88% for pharmaceutical formulations. Overall, the results indicate satisfactory precision under the investigated experimental conditions.

Statistical verification of accuracy using student’s t-test

Student’s t-test was applied to determine whether the mean recoveries obtained with the evaluated chemometric models differed significantly from the theoretical 100% value [35]. The statistical results are compiled in Tables S4 and S5. In every case, the calculated t-value remained below its respective critical value (t₀.₀₂₅,₁₅ = 2.131 and t₀.₀₁,₁₅ = 2.947), demonstrating that none of the observed recoveries differed significantly from 100% at the 95% confidence level.

For the pharmaceutical formulation, t-values of 0.78–2.04 were obtained for VEN and PIR, whereas the plasma experiments gave values between 0.86 and 2.04. Thus, the recovery data from all investigated models showed no statistically significant bias relative to the theoretical 100% value.

One-way ANOVA

By using one-way ANOVA, the six models’ quantitative findings were compared to those of the reference method [36] at the 95% confidence level to determine their statistical equivalence. Pharmaceutical formulation and plasma sample data are in Tables S6 and S7.

F values of 1.742 for VEN and 1.964 for PIR were estimated for pharmaceutical formulations, with p-values of 0.158 and 0.103 (Table S6). VEN and PIR had F values of 2.084 and 2.261 in plasma, respectively, with p-values of 0.087 and 0.064 (Table S7). All F values were below the 95% confidence level (F₆,₂₈ = 2.445) and p-values rose above 0.05. The tested models and the reference method for simultaneous VEN and PIR assessment showed no statistically significant differences.

ANOVA showed that the tested models had slight numerical variations in RMSEP, recovery, and precision, but there was no statistical significance. In conclusion, all chemometric methods performed well for pharmaceutical formulations and spiked human plasma.

Slightly higher F values and correspondingly lower p-values were observed for plasma than for pharmaceutical formulations, indicating greater variability in the plasma results; however, these differences remained statistically nonsignificant.

The statistical findings also indicate that the different calibration strategies—including CLS, PCR, PLS, GA-PLS, FA-PLS, and MCR-ALS—all achieved comparable analytical performance from a statistical standpoint. Nevertheless, MCR-ALS showed the best overall numerical performance based on the lower residual errors, higher recoveries, and lower variability observed in the investigated datasets. This numerical advantage should not be interpreted as statistical superiority, as the ANOVA results did not demonstrate significant differences among the evaluated approaches.

Regression trueness assessed using EJCR

EJCR was used as a complementary test of regression trueness for the developed chemometric models. This analysis considers the slope and intercept together and therefore allows constant and proportional bias to be examined simultaneously. Figs. S5 and S6 show the pharmaceutical formulation and spiked human plasma 95% confidence ellipses. Ideal intercept and slope are 0 and 1 for an unbiased relationship between anticipated and reference concentrations. Model confidence regions varied greatly. CLS and PCR had the broadest ellipses, indicating greater regression coefficient uncertainty and less reference value agreement. Later models, such as PLS, GA-PLS, and FA-PLS, have narrower ellipses. This behavior is compatible with latent-variable modeling and variable-selection processes improving spectral variable handling. MCR-ALS gave the most confined confidence regions among the models examined. In both the pharmaceutical and plasma matrices, the point corresponding to intercept = 0 and slope = 1 was enclosed by the respective 95% confidence ellipse. No constant or proportionate bias was statistically obvious for MCR-ALS. This matches Tables 2, 3 and 4’s lower prediction errors and increased precision. The one-way ANOVA did not show statistically significant differences between models, but the EJCR analysis evaluated the slope and intercept jointly and considered their covariance, which gave a more useful investigation of regression trueness. The EJCR results therefore support the reliability of the developed models and identify MCR-ALS as the model providing the closest agreement with the reference concentrations under the conditions investigated.

Table 3.

Determination of VEN and PIR in pharmaceutical formulations using the investigated chemometric models and comparison with the standard addition method

Preparation		%Recovery ± %RSD (a)
		CLS	PCR	PLS	GA-PLS	FA-PLS	MCR-ALS
VEN	Direct analysis	98.42 ± 1.28	99.06 ± 1.04	99.38 ± 0.86	99.57 ± 0.72	99.76 ± 0.56	99.91 ± 0.42
	Standard addition	99.11 ± 1.02	99.42 ± 0.86	99.68 ± 0.72	99.82 ± 0.61	100.04 ± 0.46	100.12 ± 0.34
PIR	Direct analysis	98.06 ± 1.46	98.74 ± 1.18	99.14 ± 0.96	99.36 ± 0.80	99.58 ± 0.64	99.81 ± 0.48
	Standard addition	98.84 ± 1.14	99.18 ± 0.94	99.46 ± 0.80	99.71 ± 0.66	99.92 ± 0.52	100.06 ± 0.39

^a Average of six determinations

Table 4.

Recovery and precision of VEN and PIR in spiked human plasma using the investigated chemometric models

Analyte	Model	Spiked Level (µg mL⁻¹)	%Recovery ± %RSD (a)	Analyte	Model	Spiked Level (µg mL⁻¹)	%Recovery ± %RSD (a)
VEN	CLS	8	95.18 ± 2.24	PIR	CLS	2	94.04 ± 2.49
		16	95.84 ± 2.02			5	95.61 ± 2.24
		24	96.31 ± 1.86			8	95.02 ± 2.02
	PCR	8	96.24 ± 1.86		PCR	2	95.94 ± 2.04
		16	96.88 ± 1.68			5	96.42 ± 1.82
		24	97.21 ± 1.52			8	96.88 ± 1.66
	PLS	8	97.02 ± 1.52		PLS	2	96.81 ± 1.72
		16	97.64 ± 1.38			5	97.21 ± 1.54
		24	97.91 ± 1.25			8	97.58 ± 1.38
	GA-PLS	8	97.74 ± 1.25		GA-PLS	2	97.34 ± 1.46
		16	98.02 ± 1.10			5	98.76 ± 1.29
		24	98.31 ± 0.98			8	98.04 ± 1.15
	FA-PLS	8	98.12 ± 1.02		FA-PLS	2	97.82 ± 1.18
		16	98.44 ± 0.92			5	98.12 ± 1.04
		24	98.67 ± 0.84			8	98.41 ± 0.92
	MCR-ALS	8	98.42 ± 0.86		MCR-ALS	2	98.18 ± 0.99
		16	98.71 ± 0.78			5	98.52 ± 0.86
		24	98.94 ± 0.70			8	98.83 ± 0.74

^aAverage of six determinations

Overall model performance and practical significance

The investigated chemometric models were compared systematically to assess their relative analytical performance by integrating the results obtained from residual error statistics (RMSEC, RMSEP, RRMSEP, and BCRMSEP), statistical validation, and multivariate EJCR analysis. This integrated assessment provides a balanced evaluation of model performance by simultaneously considering predictive accuracy, statistical reliability, systematic bias, and regression stability.

Across both pharmaceuticals and spiked human plasma, all evaluation criteria consistently demonstrated a progressive enhancement in analytical performance according to the following sequence:

CLS < PCR < PLS < GA-PLS < FA-PLS < MCR-ALS.

Among the investigated models, CLS exhibited the lowest predictive capability because it directly relates spectral measurements to analyte concentrations without dimensionality reduction. Consequently, in highly overlapping spectral systems such as the VEN–PIR mixture, the model is particularly susceptible to multicollinearity and noise propagation, leading to comparatively larger prediction errors and broader EJCR confidence regions.

Using latent-variable approaches like PCR and PLS to reduce the spectral matrix to an orthogonal space enhanced prediction reliability. Although PCR effectively captured the dominant spectral variance, the extracted components were generated independently of analyte concentration, allowing chemically irrelevant variance to remain within the model. In contrast, PLS simultaneously maximized the covariance among analyte concentrations and spectral variables, resulting in improved quantitative accuracy and greater predictive efficiency.

Further enhancement was achieved through the incorporation of metaheuristic wavelength-selection algorithms. Both GA-PLS and FA-PLS selectively retained the most informative spectral variables while eliminating redundant wavelength regions, thereby reducing calibration and prediction errors and improving model precision. These findings demonstrate the effectiveness of evolutionary optimization strategies in decreasing spectral dimensionality while retaining quantitative prediction information.

MCR-ALS showed the best overall numerical performance among the investigated models, based on the reported RMSEC, RMSEP, recovery, and precision results. This performance is consistent with the ability of MCR-ALS to decompose the experimental spectral matrix into underlying spectral profiles and concentration contributions, which can facilitate resolution of overlapping spectral information before quantitative estimation. By directly separating the contributions of the individual analytes, MCR-ALS effectively minimizes covariance-related interference resulting from severe spectral overlap. This advantage was reflected by the smallest RMSEP, RRMSEP, and BCRMSEP values, together with the smallest EJCR confidence regions centered close to the theoretical ideal coordinate.

Despite the absence of significant differences among the tested models according to one-way ANOVA at the 95% confidence level, MCR-ALS yielded the most favorable numerical results when RMSEP, recovery, precision, and EJCR were considered together.

Practically, all investigated chemometric models demonstrated suitable analytical performance for the quantification of the studied analytes in pharmaceuticals and spiked human plasma. Nevertheless, MCR-ALS may be considered the most favorable approach among the investigated models based on its numerical performance, particularly for complex spectral matrices where extensive signal overlap is present.

Overall, the integration of advanced chemometric modeling, rational experimental design, and comprehensive statistical validation provides a robust analytical framework for multicomponent UV spectrophotometric analysis. The proposed platform shows considerable potential for broader implementation in pharmaceutical quality control, exploratory bioanalytical applications, and sustainability-oriented analytical methodologies requiring rapid, reliable, and reagent-efficient quantification of complex mixtures.

Application to pharmaceutical formulations

A commercial pharmaceutical product containing VEN and PIR was selected to examine the practical applicability of the proposed procedure. Quantification was carried out by direct assay and standard addition, and the results obtained with the evaluated chemometric models are reported in Table 3.

Assay of the pharmaceutical product

The chemometric models produced consistent assay results for the commercial formulation. For VEN, the mean recovery extended from 98.42% with CLS to 99.91% with MCR-ALS. In the case of PIR, recoveries fell between 98.06% and 99.81%, with the exact value depending on the calibration model applied. All measured recoveries satisfied the relevant pharmacopeial assay requirements, indicating that pharmaceutical quality can be quantified using the proposed method.

Precision data validated the method’s reliability. For VEN, %RSD values decreased from 1.28% for CLS to 0.42% for MCR-ALS, while PIR exhibited slightly greater variability, with %RSD values declining from 1.46% to 0.48%. The relatively higher variability observed for PIR is likely attributable to its lower concentration range together with the greater influence of spectral overlap. Nevertheless, all precision values remained within acceptable analytical limits. The gradual decrease in variability following the sequence CLS → PCR → PLS → GA-PLS → FA-PLS → MCR-ALS demonstrates improved predictive stability across the investigated chemometric models.

Standard addition studies

To further verify analytical accuracy and evaluate the potential influence of formulation excipients, standard addition experiments were performed by fortifying the pharmaceutical formulation with known concentrations of VEN and PIR (Table 3). Recoveries for VEN ranged from 99.11% to 100.12%, whereas those for PIR varied between 98.84% and 100.06% across the investigated chemometric models. The concordance between the direct and standard addition findings suggests that formulation excipients did not introduce significant interference during spectral acquisition or chemometric prediction.

In addition, standard addition experiments generally produced lower %RSD values than direct analysis, particularly for GA-PLS, FA-PLS, and MCR-ALS, demonstrating improved prediction consistency following analyte fortification. Between the models, MCR-ALS achieved the highest recoveries together with the lowest variability for both analytes, highlighting its capability to resolve overlapping spectral contributions within the pharmaceutical matrix.

Overall, these findings support the applicability of the proposed method for the concurrent determination of VEN and PIR in pharmaceuticals without requiring chromatographic separation or extensive sample preparation.

Comparative performance of chemometric models in the pharmaceutical matrix

Comparing chemometric models in the pharmaceutical matrix showed that model sophistication improved analytical performance. Direct full-spectrum regression is more susceptible to spectral collinearity and instrumental noise, hence CLS had the highest prediction variability. PCR and PLS reduced spectral dimensionality through latent-variable modeling, improving predictive stability. GA-PLS and FA-PLS improved analyte classification by reducing non-informative spectral characteristics through evolutionary wavelength selection. MCR-ALS had the highest analytical stability, maximum recoveries, and lowest %RSD for direct assay and standard addition trials. MCR-ALS’ bilinear decomposition capability divides mixed spectra into component profiles before quantification, decreasing covariance-related interference from substantial overlaps.

Practical significance

The effective concurrent quantification of VEN and PIR in commercial pharmaceuticals without the necessity for chromatography illustrates the utility of the suggested method. The established analytical framework presents numerous benefits, encompassing: Precise concurrent measurement of both substances despite significant spectrum interference, insignificant disruption from pharmaceutical excipients in quantitative assessment, and high accuracy appropriate for standard pharmaceutical quality assurance procedures, in addition to Swift, economical, and ecologically viable assessment with minimal solvent usage. The aggregate findings indicate that the suggested approach offers a dependable, effective, and sustainable substitute for traditional chromatographic methods in the regular quality assessment of VEN–PIR pharmaceutical products.

Evaluation of the suggested method in plasma

The relevance of the suggested method to biologically relevant samples were assessed using blank human plasma spiked with PIR and VEN. Plasma-specific calibration models were used for quantitative measurements, with the analytical results generated by the different chemometric approaches presented in (Table 4). Owing to the intrinsic complexity of plasma, which contains endogenous proteins, metabolites, and naturally absorbing constituents that can influence spectral measurements, matrix-specific calibration models were constructed to reduce matrix-induced spectral variation and enhance predictive reliability.

Quantitative performance in spiked human plasma

Accurate quantification of both analytes was achieved despite the spectral complexity associated with the plasma matrix. Depending on the chemometric model and concentration level, mean recoveries ranged from 95.18 to 98.94% for VEN and from 94.04 to 98.83% for PIR. These recovery values indicate satisfactory quantitative performance under the investigated experimental conditions and indicate that the optimized protein precipitation procedure reduced interference from endogenous plasma components.

The developed models also exhibited generally acceptable precision throughout the investigated concentration ranges. The %RSD values varied between 0.70 and 2.24% for VEN and between 0.74 and 2.49% for PIR. MCR-ALS consistently gave the highest recovery values together with the lowest prediction variability among the investigated models.

Preliminary investigation of plasma matrix effects

The impact of remaining plasma components on analytical outcomes was further investigated using the optimized MCR-ALS model. Blank human plasma specimens were exposed to the extraction procedure, fortified after extraction with known concentrations of PIR and VEN, and subsequently compared with neat standard solutions prepared at identical concentration levels. The calculated matrix factors (MF) and corresponding matrix effects (%) are summarized in (Table S8).

For PIR, MF values ranged from 0.980 to 0.993, while VEN exhibited values between 0.984 and 0.992. These corresponded to matrix effects of − 2.0 to − 0.7% for PIR and − 1.6 to − 0.8% for VEN. The consistently small negative values indicate only slight signal suppression following protein precipitation, with all matrix effects remaining within ± 2%. These findings indicate that the optimized extraction procedure limited the contribution of residual endogenous plasma constituents under the investigated conditions.

It should be noted that this investigation was performed exclusively with laboratory-prepared post-extraction spiked plasma samples and should consequently be interpreted as a preliminary, proof-of-concept evaluation of matrix influence instead of a comprehensive regulatory bioanalytical matrix-effect assessment. Consequently, the results provide a preliminary indication that the proposed extraction protocol limited endogenous spectral interference under the investigated experimental conditions. Nevertheless, comprehensive bioanalytical validation involving incurred clinical specimens, plasma obtained from multiple donors, evaluation of metabolite-related interference, and broader regulatory studies will be necessary before extending the method to routine clinical use or pharmacokinetic investigations.

Practical utility and analytical applicability

The developed spectrophotometric–chemometric platform is not designed to substitute highly sensitive chromatographic methodologies for trace-level bioanalysis or detailed pharmacokinetic studies. Instead, it offers a rapid, economical, and environmentally sustainable alternative for plasma analysis when analytes are present at moderate concentrations. The analytical workflow relies on simple sample preparation and direct spectral acquisition without chromatographic separation, thereby markedly decreasing analysis time, solvent usage, operational expenses, and laboratory waste.

The practical applicability of the proposed approach is corroborated by the validated plasma calibration ranges together with the documented pharmacokinetic profiles of the investigated compounds. For PIR, the validated working interval (1–9 µg/mL) fully covers the reported maximum plasma concentration (Cmax = 1812.00 ± 80.76 ng/mL; approximately 1.81 µg/mL) [37], confirming that the method is appropriate for quantification across clinically relevant plasma levels. Conversely, the reported Cmax of VEN (1530.61 ± 40.65 ng/mL) lies below the validated analytical range [38]. Therefore, the proposed procedure may be applicable to samples containing elevated VEN concentrations, including samples subjected to suitable preconcentration, rather than being considered a routine therapeutic drug-monitoring method at conventional clinical concentrations.

The successful application of the external validation procedure for the determination of both analytes in plasma demonstrates shows the chemometric models’ prediction power under the tested matrix conditions. These findings reinforce the preliminary analytical utility of the developed methodology for spiked-plasma analysis in addition to pharmaceutical quality control, while comprehensive bioanalytical validation would be required for broader clinical or pharmacokinetic applications.

Comparative benchmarking of sustainability performance relative to the reported method

Beyond analytical validation, the proposed methodology was subjected to a comparative sustainability evaluation against the only previously reported analytical procedure [15]. To provide a comprehensive and balanced assessment, multiple complementary sustainability metrics were applied instead of relying on a single evaluation tool. This integrated benchmarking strategy enabled the assessment of diverse dimensions of analytical sustainability, including ecological effects, practicality, methodological innovation, overall sustainability performance, and potential contributions of the suggested method toward the United Nations Sustainable Development Goals (UN-SDGs) [39–51].

Environmental performance was assessed using the NEMI [52], the ComplexGAPI [53], and the AGREE [54]. Practical applicability was evaluated using the BAGI [55], methodological innovation was assessed using the VIGI [56], holistic sustainability was examined using the RGB12 [57] and the NQS Index (Fig. S7) [58], while broader sustainability alignment was evaluated using the SAMI [59].

The comparative sustainability assessment is summarized in (Tables S9–11). According to the evaluated sustainability metrics, the proposed spectrophotometric–chemometric methodology had better sustainability than RP-HPLC. From an environmental perspective, the proposed approach satisfied all NEMI criteria, achieved a higher AGREE score (0.76 vs. 0.52), and produced a markedly lower CFA value (0.014 vs. 0.066 kg CO₂-eq per analysis). In addition, the ComplexGAPI pictogram reflected a greener analytical workflow.

According to the evaluated practical applicability criteria, the proposed methodology also showed a more favorable profile. A higher BAGI score (80 vs. 70) was obtained, reflecting simpler instrumentation, lower solvent consumption, and shorter analysis time. The VIGI assessment also assigned a greater innovation score to the proposed method (80 vs. 65). Holistic sustainability evaluation further supported these findings, with higher RGB12 and NQS scores of 88.2 and 87, respectively, compared with 69.6 and 49 for the reported RP-HPLC procedure. Likewise, SAMI yielded a higher score for the proposed methodology (81 vs. 29), indicating stronger alignment with the selected UN-SDG-related criteria within the SAMI framework. This score represents potential contributions inferred from the evaluated analytical and sustainability characteristics rather than experimentally demonstrated societal outcomes.

Comprehensive descriptions of the individual assessment metrics, together with the corresponding scoring procedures, criterion-specific evaluations, calculation details, and graphical outputs, are provided in Supplementary Section S1.

Collectively, these benchmarking results indicate a more favorable sustainability profile for the proposed analytical strategy according to the evaluated criteria, including environmental impact and operational practicality, relative to the reported RP-HPLC method. The close agreement among NEMI, ComplexGAPI, AGREE, CFA, BAGI, VIGI, RGB12, NQS, and SAMI provides complementary evidence across multiple sustainability dimensions rather than establishing universal superiority of one analytical procedure over the other.

Limitations of the present study and future research directions

The present work demonstrates satisfactory concurrent determination of PIR and VEN in pharmaceuticals and spiked human plasma; however, the plasma experiments do not establish full clinical applicability. In particular, the bioanalytical part of the study was restricted to plasma samples prepared by laboratory spiking. The obtained results therefore demonstrate the feasibility of applying the proposed platform to a biological matrix, but they do not constitute a complete bioanalytical validation. Consequently, the method should not yet be considered suitable for therapeutic drug monitoring, pharmacokinetic investigations, or clinical decision-making. The present work also did not address several aspects typically included in comprehensive bioanalytical validation, including analysis of incurred clinical specimens, assessment of donor-to-donor variability, investigation of metabolite interference, evaluation of long-term analyte stability under biological storage conditions, and extensive matrix-effect studies involving plasma from multiple sources. These investigations were deliberately excluded because the principal objective was to establish the feasibility of integrating green UV spectrophotometry via chemometrics for the quantification of PIR and VEN in biologically relevant matrices.

Notwithstanding these limitations, the proposed analytical strategy provides a rapid, economical, and environmentally sustainable approach for the quantification of both analytes in pharmaceuticals and spiked plasma samples, while reducing solvent consumption, instrumental requirements, analytical time, and operational cost. Future studies should extend this work through full bioanalytical validation using incurred clinical samples collected from multiple donors, comprehensive evaluation of metabolite interference and inter-matrix variability, detailed stability investigations under clinically relevant storage and handling conditions, and systematic comparison with established chromatographic reference methods. These efforts would further clarify the analytical capabilities and potential clinical utility of the proposed methodology while expanding its applicability to broader bioanalytical investigations.

Conclusion

A sustainable UV-based multivariate analytical method was established for the simultaneous quantification of VEN and PIR in pharmaceuticals and as a proof-of-concept application to spiked human plasma. By integrating conventional UV spectrophotometry with established multivariate calibration techniques, the proposed approach successfully resolved extensively overlapping absorption spectra without the need for chromatographic separation. Between the assessed chemometric models, MCR-ALS showed the best overall numerical performance, although all investigated models provided satisfactory analytical results within their validated concentration ranges. The developed methodology demonstrated appropriate accuracy, precision, sensitivity, and reproducibility for its intended analytical applications.

Experimental efficiency was further enhanced through the combined implementation of Brereton’s multilevel calibration design and MMD, which enabled representative calibration and validation datasets to be generated using a reduced number of experimental mixtures. The environmental and practical attributes of the proposed method were comprehensively benchmarked against the previously reported chromatographic procedure using multiple complementary sustainability assessment tools. The collective outcomes of these assessments indicated a more favorable sustainability profile for the proposed methodology according to the evaluated criteria, including lower solvent consumption, reduced instrumental complexity, and lower associated energy and waste requirements, while maintaining satisfactory analytical performance.

The feasibility of extending the proposed methodology to a biological matrix was explored through the analysis of spiked human plasma following protein precipitation and application of plasma-specific chemometric calibration models. Nevertheless, this inquiry need to be considered a proof-of-concept study instead of a comprehensive bioanalytical validation. Further work involving incurred clinical specimens, evaluation of inter-donor variability, assessment of metabolite interference, comprehensive stability studies, and broader investigations of matrix effects will be necessary before the method can be considered for routine clinical analysis, therapeutic drug monitoring, or pharmacokinetic applications.

Overall, coupling well-established chemometric techniques with a rationally planned experimental design and an extensive sustainability assessment yielded a practical analytical platform for the concurrent quantification of PIR and VEN in pharmaceutical dosage forms, with preliminary proof-of-concept applicability demonstrated in spiked human plasma. This approach represents an economical, practical, and environmentally conscious solution for pharmaceutical quality control and initial matrix examinations, while also establishing a basis for the continued advancement of sustainability-driven analytical methodologies.

Abbreviations

AGREE

Analytical GREEnness Metric

ANOVA

Analysis of Variance

BAGI

Blue Applicability Grade Index

BCRMSEP

Bias-Corrected Root Mean Square Error of Prediction

CFA

Carbon Footprint Assessment

CLS

Classical Least Squares

ComplexGAPI

Complex Green Analytical Procedure Index

EJCR

Elliptical Joint Confidence Region

FA-PLS

Firefly Algorithm-assisted Partial Least Squares

GA-PLS

Genetic Algorithm-assisted Partial Least Squares

GSST

Green Solvent Selection Tool

ICH

International Council for Harmonisation of Technical Requirements for Pharmaceuticals for Human Use

LOD

Limit of Detection

LOQ

Limit of Quantification

LV

Latent Variable

MATLAB

Matrix Laboratory

MCR-ALS

Multivariate Curve Resolution–Alternating Least Squares

MMD

Maximin Distance Design

NAS

Net Analyte Signal

NEMI

National Environmental Methods Index

NQS

Need–Quality–Sustainability Index

PCA

Principal Component Analysis

PCR

Principal Component Regression

PIR

Piroxicam

PLS

Partial Least Squares

RGB12

Red–Green–Blue Twelve-Parameter Sustainability Algorithm

RMSEC

Root Mean Square Error of Calibration

RMSECV

Root Mean Square Error of Cross-Validation

RMSEP

Root Mean Square Error of Prediction

RRMSEP

Relative Root Mean Square Error of Prediction

%RSD

Percent Relative Standard Deviation

SAMI

Sustainability Assessment of Analytical Methods Index

SDAGI

Spider Diagram for Assessment of the Greenness Index

SEC

Standard Error of Calibration

VEN

Venlafaxine Hydrochloride

VIGI

Violet Innovation Grade Index

Authors’ contributions

Bshra A. Alsfouk: Conceptualization, methodology, supervision, investigation, formal analysis, data interpretation, writing – original draft, writing – review & editing.Lateefa A. Al- Khatee and Omkulthom Al kamaly: Methodology, investigation, visualization, figures preparation, formal analysis, writing – original draft, writing – review & editing.Mahmoud A. Tantawy: Methodology, writing – original draft, writing – review & editing. Moayad M. Khashoqji: Methodology, investigation, visualization, figures preparationMichael K. Halim: Investigation, sample preparation, analytical validation, formal analysis, data curation, writing – original draft, writing – review & editing. Sona S. Barghash: writing – review & editing, figures preparation, investigation, validationAhmed Emad. F. Abbas: Conceptualization, methodology, experimental design, investigation, formal analysis, chemometric modeling, visualization, supervision, writing – original draft, writing – review & editing.All authors contributed to data analysis, manuscript revision, and final approval of the submitted version.

Funding

This research was funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2026R142), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Data availability

The corresponding author will provide the datasets created and/or analyzed during the current study upon reasonable request.

Declarations

Ethics approval and consent to participate

No human participants were enlisted or directly engaged in this research. and no clinical procedures or newly collected human biological samples were undertaken. For the plasma studies, drug-free human plasma was acquired as a commercially available, completely anonymized biological substance from the Egyptian Holding Company for Biological Products and Vaccines (VACSERA, Giza, Egypt). Because the plasma was obtained anonymously for analytical method development and was not collected specifically for the present investigation, the study falls outside the institutional ethical approval and informed consent requirements applicable to research involving identifiable human participants, as stipulated by Article 3 of the Egyptian Clinical Research Law No. 214 of 2020 and the relevant VACSERA policy governing secondary use of biological materials. The experimental work was nevertheless performed in accordance with internationally accepted ethical principles, including those set out in the Declaration of Helsinki and its subsequent amendments.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.