Thermal Degradation of β-Carotene from Macauba Palm: Mathematical Modeling and Parameter Estimation

Worldwide, there is a current need for new sources of vegetable oils. The natural content of total carotenoids in Acrocomia aculeata palm oil (up to 378 μg.g−1) surpasses that of many other tropical fruits, making it one of its main compositional characteristics. As far as can be verified, there is no available information on the degradation kinetics of carotenoids for A. aculeata oil, which is required to describe reaction rates and to predict changes that can occur during food processing. The present study considered prediction abilities that have emerged with the use of specific kinetic data and procedures to understand thermal processing better, as an essential unity operation. Two kinetic mechanisms were proposed to describe the overall thermal degradation of carotenoids in the oil; the first one consists of three reaction steps while the other presents only one-step reaction. Mass balance equations were numerically solved by a Backward Differentiation Formula technique. The kinetic parameters from both models were estimated through a hybrid optimisation method using the Particle Swarm Optimization and the Gauss-Newton method, followed by statistical analyses. The model with more than one reaction was shown to be overparameterized and was discarded. The model with a single reaction was highly suited to handle the experimental data available, and the dependency of its rate constant on temperature was expressed according to Arrhenius law. As far we know, this is the first time the kinetics of carotenoids thermal degradation in A. aculeata oil is investigated through modelling


Introduction
From the perspective of extractive practices concerning oil crops, Acrocomia aculeata, known as macauba, is a palm tree widely dispersed in Brazil. Over the last few years, the food and chemical industries have been responsible for most of the growing demand for vegetable oils. In this context, the interest in macauba as a food product has increased due to factors such as the nutritional quality of the oils extracted from its edible parts. The industrial interest in macauba has also involved full use of its fruit to generate co-products with added value. The fruit mesocarp and kernel together correspond to approximately 47% (on dry basis) of the total fruit weight. Noticeably, the mesocarp contributes to around 60% (on dry basis) of the total oil content, with a predominance of oleic ω-9 (53%) and linoleic ω-6 (18%) acids. The kernel oil is predominantly saturated, with around 40% of lauric acid. An adult Acrocomia aculeata palm fructifies almost throughout the year with productivity ranging from 4 to 6 tonnes of esculent oil per hectare. The crop has, accordingly, a similar productive potential to Elaeis guineensis, which is among the highest oil-yielding plants in the world (Evaristo et al., 2016). Considering that high yield has been an essential characteristic for the selection of commercial harvests, it has potential as an alternative oil crop that deserves further investigation (Rodriguez-Amaya, Kimura, Godoy & Amaya-Farfan, 2008). Regarding the oil extracted from the fruit mesocarp, it contains no antinutritional factors and contains up to 378 mg.kg −1 of total carotenoids, mainly β-Carotene (Nunes, Favaro, Galvani & Miranda, 2015). The carotenoids are essential pigments in fruits, these tetraterpenes (C40) synthesised by plants are secondary metabolites, necessary for photosynthesis and to prevent photo-oxidation induced by light intensities.
These functions are a consequence of the light-absorbing properties of their polyene chromophore (Oloo, Shitandi, Mahungu, Malinga & Ogata, 2014;Rodriguez-Amaya, Rodriguez & Amaya-Farfan, 2006;Schieber & Carle, 2005). In general, carotenoids naturally exist as all-trans form. However, isomerisation of all-trans-carotenoids to cis forms is one of the major reactions of the compound's degradations. The critical step for losses of the component in vegetable oils remains related to the exposure to high temperature, light or pro-oxidant molecules. Indeed, the elevation of temperature during thermal treatments has been shown to dramatically increase corresponding degradation reactions rates (Achir, Randrianatoandro, Bohuon, Laffargue & Avallone, 2010;Sampaio et al., 2013). The intensity of the thermal treatment is a critical factor that has to be controlled to increase carotenoid retention in the macauba mesocarp oil. As an essential unit operation, the thermal processing influences both carotenoids bioaccessibility and the health-related attributes of vegetable oils (Nunes et al., 2015;Palmero et al., 2013;Rodriguez-Amaya et al., 2006). Along with carotenoids products' identification, kinetic data become necessary to predict carotene loss on thermal degradation accurately. Kinetic evaluation is therefore required to derive necessary kinetic information for a system to describe the reaction rate as a function of experimental variables also predicting changes in a particular food system during processing. In general, most of the studies in real food report a first-order reaction on the concentration of transβ-carotene, in different systems, at different processing temperatures. Although zero-order equations have also been verified, the use of a firstorder kinetic is realistic in most cases. As far as we have knowledge there is no available information about thermal degradation kinetic modelling of neither β-carotene nor any other carotenoids from macauba oil (Achir, Penicaud, Avallone & Bohuon, 2011;Ahmed, Shivhare & Sandhu, 2002;Knockaert et al., 2012;Penicaud, Achir, Dhuique-Mayer, Dornier & Bohuon, 2011). Some studies in nonpolar solvents tested reaction orders superior to one for trans-β-carotene degradation and found a better fit of experimental data by linearisation or nonlinear regression methods. The superior orders may be explained by the competition with isomerisation reactions, which are also of importance in vegetable oils. Most of the kinetic models used to describe trans-β-carotene degradation are single response kinetic models. However, as the compound is supposed to generate various degradation products, the original reaction scheme can be complex to involve complex dynamics (Achir et al., 2010;Penicaud et al., 2011;Sampaio et al., 2013). Regarding the estimation of kinetic parameters for trans-β-carotene degradation, the rate constants k (s −1 ) can vary ranging from 0.00018 (120 o C) to 0.0015 (180 o C). The apparent activation energy E a (kJ.mol −1 ) tends to range from 80 to 110 (Achir et al., 2010;Dhuique-Mayer et al., 2007;Henry, Catignani & Schwartz, 1998;Sampaio et al., 2013). The objective of the present study was to predict the thermal degradation of all-trans-β-carotene in Acrocomia aculeata oil through mathemat-IJFS April 2021 Volume 10 pages 161-172 ical modelling, emphasizing the importance of numerical predictions for practical applications. This study is an important step to comprehend better the kinetics of thermal degradation of macauba oil, which is a promising source of highquality raw materials.

Crude Oil: Acrocomia aculeata
Acrocomia aculeata fruit was collected from native palms with a maximum of five days after the fall in the Federal University of Minas Gerais -UFMG, located in the metropolitan region of Belo Horizonte, Minas Gerais, Brazil. The mesocarp and kernel portions were promptly separated from the fruit. Before the oil extraction, the mesocarp and kernel portions were thawed, airdried at 60 o C for 48 hours and comminuted in an electric grinder coupled to a stainless steel cup (Goula, 2013;Pimenta, 2010). For the present study, the samples consisted of edible oil mechanically obtained from the macauba mesocarp by a continuously operated Expeller ® press, at 34 o C.
Amber glass vials (15 mL) were filled to the maximum working volume with the samples, minimising the impact of light and oxygen intrusion by reducing the volume of headspace. Samples were stored at freezing temperature (18 o C) until the analysis to minimise possible rates and extents of enzymatic lipolysis, also concerning potential losses of antioxidants (Koidis & Boskou, 2015;Parducci & Fennema, 1978).

Carotenoid Determination
The HPLC analyses of carotenoids were carried out on a Shimadzu system (Shimadzu, Japan) equipped with a vacuum degasser, a quaternary pump and an autosampler (SIL-20A HT). A UV-Visible photodiode array detector (SPD-M20A) was set in the range of 190 -800 nm to analyse the chromatograms. Once β-carotene represent around 90% of the total carotenoids content in the A. aculeata mesocarp oil (Coimbra & Jorge, 2012;Nunes et al., 2015), peaks were detected at 455 nm, and the results were expressed as alltrans-β-carotene. The separation was achieved at 30 o C using a normal phase column (Phenomenex Luna Silica (2) 100A Si: 250 mm × 4.6 mm i.d., 5 µm particle size) prior equilibrated with a flow of 0.1 mL.min −1 . The mobile phase was n-hexane/isopropyl alcohol (97.0:3.0 v/v), the flow rate was maintained at 1.0 mL.min −1 , and the elution remained isocratic until 26 min. After every 10 injections of 20µL, the column was reactivated with a solution of 10% isopropyl alcohol in n-hexane (v/v). The carotenoids were identified for each experimental condition by the combined use of their relative retention times and previously published UV/Vis spectra (Panfili, Fratianni & Irano, 2004;Rodriguez-Amaya, Kimura et al., 2004

Thermal Degradation of all-trans-β-Carotene
For all the experiments, the initial carotenoid concentration (determined according to item 2.2) was applied. 0.20 mL of oil (ρ-macauba oil = 925.6 g L −1 ) was used.

Model Development and Numerical Approach
A first-order kinetic mechanism (Mechanism I) with three irreversible reaction steps, represented in Equations (1) to (3), was initially chosen to be tested if it can represent well the thermal degradation of β-carotene in macauba oil.
In the above equations, A represents the alltrans-β-carotene, B represents the oxidation and cleavage products (OCP), C represents ciscarotenoids (mainly, 9 or 13 -cis-β-carotene). Also, k AB and k CB correspond to the rate constants for the thermal degradation of all-trans and cis-β-carotene, respectively. k AC corresponds to the carotene rate constant for the isomerisation of trans-β-carotene.
Assuming an isothermal, the material balances for all species involved in Mechanism I are represented by: A second mechanism (Mechanism II) was also considered to be a candidate mechanism able to predict well the experimental data available from the thermal degradation of β-carotene. This second mechanism is a simplified version of the first one, and it is shown in Equation (7).
In the above equation, A * represents the sum of all-trans and cis-β-carotene, and k A * B correspond to the rate constant for the thermal degradation of β-carotene. Once more, assuming an isothermal, the material balances for all species involved in Mechanism II are represented by: The ordinary differential equations resultant from the material balances (Equations (4) to (6) and (8) and (9)) were then solved using the BDF (backward differentiation formula) technique, as programmed in the DASSL code (Petzold, 1982). Parameter estimation was performed with the package ESTIMA, implemented in Fortran, using a hybrid optimisation method PSO (particle swarm optimisation) and Gauss-Newton algorithms (Brandao, Oechsler, Gomes, Souza & Pinto, 2018; Schwaab, Biscaia, Monteiro & Pinto, 2008). Five hundred particles were used, and two thousand iterations were performed with a numerical tolerance of 0.0001 for the objective function. Besides, a confidence level of 95% was considered, and all the parameters from Mechanisms I and II were estimated in their absolute form. The known weighted leastsquares function was used as the objective function in the present research, and it was defined as follows: F obj = (y e − y m (x m , θ)) T V ( y − 1)(y e − y m (x m , θ)) (10) In the above equation, y e and y m are the vectors for the measured and predicted dependent variables, respectively; V y is the covariance matrix of the measured outputs (assumed to be diagonal); and x m and υ are the vectors of the measured independent variables and model parameters, respectively. Experimental variances were obtained through replicates and are illustrated in the following sections. The experimental data used to estimate the parameters was the amounts of all-trans-βcarotene concentration presented at certain times  Figure 1 shows the parameters estimation procedure performed in this study.

The task and project brief
The initial concentration of carotenoids was determined as 224.1 mg for each Kg of macauba oil. Regarding the estimation procedure, it was conducted for each experimental condition separately, as shown in Figure 1. Table 1 shows the results for the objective function, the estimated parameter and respective uncertainty obtained when model derived from Mechanism I was considered. The experimental data and the model predictions within their uncertainties are shown in Figure 2. From Table 1, except for reaction temperatures of 140 and 150 o C, it is possible to observe that parameters uncertainties could not be calculated. Although all objective functions values were between the lower and upper limits of the χ2-distribution, the first model was not able to fit the experimental data available well since the model is over parametrised. At experimental conditions 140 and 150 o C, practically all parameters can assume zero value, which shows they are not significative and, consequently, they can receive any value that won't change model prediction including zero or even negative values. Figure 2 shows that predicted and experimental profiles for the ratio A/A 0 during the reaction are in good agreement which was expected since the model has more parameters it requires, so more than one parameter combination can provide apparent good results. Since the parameters uncertainties for experiments at temperatures 100, 110 and 130 o C could not be obtained through the parameter estimation procedure, as was done for the experiments done at 140 and 150 o C, at least the parameters confidence regions for these three experimental conditions could be determined, providing an idea about the values the parameters k AB , k AC and k CB can assume. So, Figure 3 shows the parameters confidence regions for experiments conducted at 100, 110 and 130 o C using the model derived from Mechanism I. In all confidence regions the estimated parameter values, presented in Table 1, are highlighted. From Figure 3, it is possible to observe that, in all confidence regions, at least one kinetic parameter can assume zero value or even negative values. The regions are not well defined since parameter uncertainties are significantly high, also suggesting that the confidence region of parameters uncertainties become open (Schwaab et al., 2008), indicating estimability problems. The estimation of all parameter uncertainties simultaneously is unfeasible for these experimental conditions using Mechanism I. Based on what was said, the model derived from Mechanism I can be unconsidered as a candidate model to represent the experimental data available of thermal degradation of macauba oil. The final objective function values for the five estimations done were between the lower and upper limits of the χ2-distribution. Therefore, according to the χ2 statistical test, the proposed model fitted experimental data well, and was suitable to predict the experimental data available. Besides, all parameters uncertainties could be calculated, and they are smaller than the estimated parameters.     (11). This effect of temperature on the kinetic constant k A * B is shown in Figure 5. It can be observed that the estimated kinetic parameters follow the Arrhenius correlation very closely since the rate constants increase with the increase of reaction temperature. Linear regression, with a coefficient of determination (R 2 ) equals to 0.9891, was carried out to calculate the activation energy (E a ) and pre-exponential factor (A) for k A * B constant shown in Table 2. The obtained fits can also be observed in Figure 5.
In the presented Equation (11), T is the reaction temperature in Kelvin (K), A is the preexponential factor expressed in the same units as k A * B and R is the universal gas constant (8.314 J mol −1 K −1 ).
For total all-trans-β-carotene degradation, a preexponential factor of 1.58·10 8 min −1 and an activation energy of 83.8 kJ mol −1 K −1 were found. kJ.mol −1 . On the other hand, Chen and Huang (1998) reported an activation energy equal to 39 kJ/mol for total β-carotene degradation of an all-trans-β-carotene standard dissolved in hexane. Depending on the carotenoid source, as well as on the processing conditions and the reaction medium, a range from 20 to 171 kJ/mol has been reported for the activation energy related to the thermal degradation of β-carotene Achir et al., 2010;Penicaud et al., 2011;Sampaio et al., 2013). To the best of our knowledge, the kinetics of the thermal degradation of β-carotene in Acrocomia aculeata oil has not been investigated yet. Therefore, this result shows the thermal degradation kinetics for the all-trans-β-carotene in the oil agrees with those of previous studies carried out on other diverse sources of vegetable oils, since the activation energy estimated in the present study is indeed IJFS April 2021 Volume 10 pages 161-172

Conclusion
The thermal degradation predicted for the of all-trans-β-carotene in the Acrocomia aculeata oil was verified through mathematical modelling. On the one hand, the use of a kinetic mechanism with more than one reaction step is unnecessary since it results in an overparameterized model. On the other hand, the model developed with just one reaction channel can simulate all the experimental conditions directly and effectively, it adequately fitted the available data within the experimental errors. The activation energy and the pre-exponential factor calculated from the Arrhenius clearly shows the dependence of the reaction rate constant k A * B on temperat-ure. The developed model with a unique reaction can be suggested to be used as an effective tool to optimise process conditions not only in laboratory but also on industrial scale. This study has provided further information enabling better comprehension of the kinetics of thermal degradation of macauba oil, which is a promising source of high-quality raw materials. carry out this research.