## 1 Introduction  
Although pharmaceuticals are essential for treating various medical conditions, water contamination by pharmaceutical residues, such as paracetamol (PAR), an analgesic [1], and metronidazoleazole (MTZ), widely used as an antibiotic and antiparasitic [2], has become a growing environmental and public health concern. These compounds are increasingly common in water bodies due to inadequate discharge of domestic, hospital, and industrial effluents and the lack of effective wastewater treatment [3, 4]. Even at low concentrations, both PAR and MTZ can cause adverse impacts on aquatic ecosystems and pose potential risks to human health due to their presence in an unmetabolized form and challenges in wastewater management [5, 6]. Therefore, water treatment strategies must be developed and improved to address the challenges posed by the presence of these emerging contaminants, ensuring water security and public health. Regulatory agencies have proposed threshold values for the presence of pharmaceutical residues in aquatic environments. For instance, the European Medicines Agency (EMA) and various national agencies have suggested environmental quality standards (EQS) in the range of 0.01 to 10 \(\upmu\)g/L for pharmaceutical residues [7]. Specifically, proposed maximum concentrations for PAR and MTZ in surface or drinking water are typically below 1-2 \(\upmu\)g/L, although exact values may vary depending on the country or guideline [8]. In the United States, the Environmental Protection Agency (EPA) includes several pharmaceutical compounds in its Contaminant Candidate List (CCL), and although not all have enforceable limits, both are monitored due to their potential environmental relevance [9]. In contrast, Brazil currently does not establish maximum permitted concentrations for these specific pharmaceutical compounds in its water quality regulations, such as CONAMA Resolution No. 357/2005, highlighting a regulatory gap that further underscores the need for improved monitoring and treatment approaches. For many years, the predominant approach in water and effluent analysis and treatment focused on individual compounds, treating each contaminant separately [2; 10; 11; 12]. This results in a limited understanding of the potential synergistic or antagonistic effects among different substances present in the water [13]. As a result, treatment methods tended to be specific to a single contaminant, neglecting the complexity of contamination by multiple pollutants. The simultaneous removal of contaminants emerges as a pressing need due to the complexity of water contamination by substances from various sources [14; 15; 16]. However, this approach faces significant technical challenges since these contaminants can interact with each other, affecting the efficacy of removal processes and potentially causing adverse impacts. Furthermore, compliance with regulations and standards regarding concentration limits in water requires the development of more efficient and economical treatment technologies capable of dealing with this complexity in an integrated manner [14; 17]. There are some studies dedicated to developing processes aimed at the simultaneous removal of contaminants in water [18; 19; 20; 21; 22]. Among these, those exploring the principles of adsorption stand out [23; 24; 25; 26; 27]. Being a technique with benefits such as high efficiency and easy implementation in treatment processes, adsorption is widely employed for the removal of different contaminants, applicable in batch processes suitable for treating small water volumes or fixed-bed columns preferable for larger volumes due to their efficiency and continuous treatment capability [28; 29; 30]. Additionally, the use of conventional adsorbents like activated carbon offers a unique combination of properties, including large surface area and porosity, favoring the effective adsorption of pollutants [31]. The adsorption models can be characterized as an inverse problem since they are non-linear equations.