# Normalized vs factorized spectra: comparative spectrophotometric approaches for enhancing the resolution of zero-order absorption spectra of olanzapine and fluoxetine in mixtures

## TL;DR

This paper introduces new spectrophotometric methods to accurately determine olanzapine and fluoxetine in mixtures without prior separation.

## Contribution

The novel approach uses normalized and factorized spectra with mathematical techniques to resolve overlapping drug spectra in their original zero-order form.

## Key findings

- The methods showed linearity for olanzapine (1.5–14 µg/mL) and fluoxetine (3.5–35 µg/mL) with high sensitivity.
- The methods were successfully applied to synthetic mixtures and pharmaceutical formulations with excellent accuracy and precision.
- Green Analytical Chemistry principles confirmed the environmental and practical merits of the proposed methods.

## Abstract

Simple, accurate, and precise UV spectrophotometric methods were developed and validated for the simultaneous determination of olanzapine (OLA) and fluoxetine (FLU) in bulk powders and combined pharmaceutical formulations without prior separation steps. The methods employed advanced mathematical manipulation techniques using both the Constant Multiplication method (CM) coupled with Spectrum Subtraction (SS) and the Factorized Zero-Order method (FZM), which was also coupled with SS. These approaches utilized normalized and factorized spectra, respectively, to resolve the overlapping spectral profiles of the two drugs. Unlike conventional methods reported in the literature, such as derivative spectrophotometry, our approach enabled the extraction of each component in its original zero-order form, exhibiting spectral features identical to those of pure standards, which enhances accuracy and precision. A direct comparative evaluation of CM–SS and FZM–SS is presented, highlighting their respective analytical merits. The proposed methods demonstrated linearity over concentration ranges of 1.5–14 µg/mL and 3.5–35 µg/mL for OLA and FLU, respectively, with limits of detection of 0.15 µg/mL (OLA) and 0.38 µg/mL (FLU), and limits of quantification of 0.47 µg/mL (OLA) and 1.15 µg/mL (FLU), confirming the high sensitivity of the methods. They were successfully applied for the simultaneous determination of OLA and FLU in synthetic mixtures and in their combined dosage form with excellent accuracy and precision. Furthermore, the environmental and practical merits of the methods were evaluated through the principles of Green Analytical Chemistry (GAC) and White Analytical Chemistry (WAC). The greenness of the proposed spectrophotometric method was evaluated by the Analytical Greenness Metric (AGREE). The overall method performance, including validation efficiency, ecological impact, and practicality, was systematically evaluated using the RGB-based WAC approach.

The online version contains supplementary material available at 10.1186/s13065-026-01722-3.