## 1 Introduction
Many of these production plants are generally devoted to the SFE process involving a preliminary Sc-CO\({}_{2}\) extraction of natural products, leading to the recovery of high-value products which provide interesting options for their use in the nutraceutical and functional food industry [20]. In the last decades, the search for bioactive compounds or \"target molecules\" from natural sources or their by-products has become the most important application of SFE technology [21; 22; 23; 24]. As reported in the literature [21], the bioactivities from natural compounds extracted by SFE from 2010 to 2015 were mainly the antioxidant (41%), antitumor (18%), and antibacterial (10%) ones. As widely reported [25; 26] SFE is a green technology that shows immediate advantages over traditional extraction techniques: (i) it is a flexible process due to the possibility of continuous modulation of the solvent power/selectivity of the supercritical fluid (SF); (ii) it allows the removal of polluting organic solvents as well as that of the expensive extract post-processing used for solvent elimination, thus ensuring a safe separation process both for human health and the environment. Based on the work of [PERSON] et al. [26] and references within, it is possible to affirm that the last decades have seen great advances; among them full characterization and quantification of supercritical extracts, assessment of kinetic and equilibrium aspects, and phenomenological modelling and optimization of operating conditions. In particular, many authors have studied the dependence of the solubility of different carotenoids in supercritical CO\({}_{2}\) with temperature and pressure [26; 27]. Most of the solubility data of these works were correlated using the semi-empirical Chrastil's model [28], which provides a proportional relationship between the solubility and density of CO\({}_{2}\)[27]. Furthermore,the majority of the SFE studies for the recovery of carotenoids have focused on tomato products and industrial tomato by-products, as they constitute a good source of lycopene, and, to a lower extent, of \(\beta\)-carotene [4]. Nonetheless, according to [PERSON] et al. [26], solute-matrix interactions can be better understood and correctly taken into account by reliable predictive models. In such a context, to better analyze the experimental results and to optimize the working parameters (temperature, pressure, etc. ), this extractive technology could greatly benefit from mathematical models that are not only suitable and reliable, but also easy to use. Different phenomena as phase equilibrium, mass transfer, and flow of Sc-CO\({}_{2}\) through packed beds, are differently involved in the kinetic models reported in the literature [29]. In many cases, the extraction of the first fraction of extracts is essentially limited by its solubility, whereas the extraction rate of the remaining fraction is limited by internal diffusion through the matrix [30]. As reported in previous papers, a simplified mathematical model was introduced to describe the time evolution of SFE of the lipid fraction from oilseeds [31; 32; 33] and microalgae [34; 35; 36]. Using [PERSON]'s equation [28], it was also possible to correlate the maximum extraction rate with both working pressures and temperatures. On this basis, a simplified method to estimate the time evolution of carotenoid extraction by Sc-CO\({}_{2}\) from two different matrices (chili peppers and tomato by-products) can be developed in order to highlight the \"matrix effect\" on the SFE of carotenoids. This new method could potentially simplify the identification of the best working conditions to promote SFE of carotenoids from these two matrices as a function of temperature, pressure, flow-rate of Sc-CO\({}_{2}\), and amount of matrix, and to reduce the load of the related experimental activity [29; 37].