Despite the significant expansion of global pistachio production, particularly in the United States, Iran, and Turkey, the processing and valorization capacities in several importing countries, including Morocco, have not progressed at the same rate. This imbalance has created a disparity between imported volumes and local processing infrastructures. In Morocco, pistachio consumption continues to increase, with most of the supply relying on imports from the United States, Turkey, and Iran (USDA, 2023). In a context marked by increasing global food demand and growing concerns regarding food quality and safety, the intersection between food security, nutrition, and sustainable agriculture has become essential for achieving global development goals. Therefore, improving storage practices and post-harvest management is now considered a major challenge to ensure the quality and safety of marketed food products (Dhineshkumar and Vigneshwaran, 2025).
       
However, major challenges persist in post-import management, quality control, storage practices and conservation conditions, all of which significantly influence product stability. Pistachios are highly sensitive to environmental factors such as humidity, temperature and oxygen exposure. Their moisture status is therefore a critical determinant of their physicochemical stability, microbiological safety and shelf life (Kader, 2006). Proper control of storage conditions is therefore essential for preserving food quality during storage. Indeed, the shelf life of fresh products depends on numerous parameters such as storage temperature, respiration rate, film gas permeability, headspace composition, product weight, as well as their interactions throughout the storage period (Hlungwani et al., 2025).
       
Moisture content and water adsorption behavior directly affect the risks of mold growth, aflatoxinss production and lipid oxidation in pistachios (Mousavi et al., 2018). Inadequate storage conditions-particularly high relative humidity in certain Moroccan warehouses or retail environments-can promote fungal proliferation and increase the likelihood of aflatoxins contamination in imported pistachios (Ernest et al., 2015).
       
Beyond fungal hazards, microbial safety also represents a serious concern. Several cases of salmonellosis have been associated with pistachio consumption (Aycicek and Yarsan, 2020, Centers for Disease Control and Prevention, 2016) and contamination by Salmonella has been confirmed through product investigations and recalls (Harris et al., 2016; Yada and Harris, 2018). Studies have demonstrated that Salmonella, Escherichia coli and Listeria monocytogenes can persist in pistachios under typical storage conditions (Kimber et al., 2012). Among these pathogens, Salmonella is considered the most heat-resistant and is therefore used as the reference microorganism for validating pathogen reduction processes in low-moisture foods, including pistachios (Lambertini et al., 2017).
       
Water activity (a_w) plays a central role in these deterioration mecWater activity (a_w) plays a central role in these deterioration mechanisms. It governs physicochemical reactions such as lipid oxidation and enzymatic activity, as well as microbial growth and survival (Mousavi et al., 2018). Consequently, understanding the hygroscopic behavior of pistachios through adsorption isotherms is essential. Sorption isotherms describe the relationship between moisture content and water activity at constant temperature, enabling prediction of product behavior under various environmental conditions. This approach helps anticipate quality losses such as texture softening, oxidative degradation, and microbial persistence. Similar observations have been reported in various agricultural and food products, where water activity and temperature significantly influence moisture dynamics and storage stability (Sharma et al., 2021; Patel et al., 2020). Moreover, studies on food and legume systems have demonstrated that sorption behavior is strongly dependent on environmental conditions and product composition (Kaur and Sandhu, 2022; Verma et al., 2019). In this perspective, primary processing technologies also play a key role in enhancing the added value of agricultural products. They contribute to reducing post-harvest losses, improving product quality, and strengthening product stability during storage and commercialization. Beyond these benefits, such technologies also support farmers and promote better valorization of agricultural resources (Mohanapriya et al., 2025).
       
Water adsorption in food systems is a complex phenomenon influenced by structural and compositional factors. To describe this behavior, numerous mathematical models have been developed and can generally be classified into theoretical, empirical and semi-empirical models. Empirical models such as Oswin, Smith and Henderson are typically based on two parameters. Semi-empirical models, notably the Peleg model (1993), involve four parameters and are widely applied due to their flexibility. Theoretical models, derived from Langmuir’s concept, include the BET (Brunauer-Emmett-Teller) and GAB (Guggenheim-Anderson-de Boer) equations, which are extensively used to model water sorption in food matrices.
       
In this context, the present study aimed to analyze the combined effects of water activity and temperature on the moisture content of pistachios (Pistacia vera L.). The objective was to establish a predictive mathematical model capable of describing moisture evolution as a function of these two parameters. Ultimately, this work seeks to contribute to the optimization of storage conditions for pistachios marketed in Morocco by improving control over their physical, chemical and microbiological stability.

The experimental study was conducted during the 2024 research period at the Laboratory of Environmental, Ecological and Agro-Industrial Engineering, Faculty of Sciences and Techniques, Université Sultan Moulay Slimane, Beni Mellal, Morocco.
 
Raw material
 
Roasted and salted pistachios (Pistacia vera L.), imported from the United States and commercialized in bulk in various retail outlets within the Fes-Meknes region (Morocco), were selected for this study. The sample analyzed was collected during the 2024 marketing campaign.
 
Experimental design
 
The equilibrium moisture content (Mc) was measured experimentally at three different temperatures (25^oC, 35^oC and 45^oC) and five distinct water activity levels (a_w = 0.0019, 0.0483, 0.3524, 0.5809, 0.8280) (Sami et al., 2026). These conditions formed a 3 x 5 factorial design, resulting in 15 trials that covered a wide range of sorption environments relevant to pistachio storage. Moisture uptake was measured using the static gravimetric method until hygroscopic equilibrium was reached (Fig 1).


       
The equilibrium moisture content (Mc) was determined using the static gravimetric method, a reference technique known for its precision, reproducibility and thermodynamic reliability in sorption studies (Yazdani et al., 2006) Spiess and Wolf, 1983). A full factorial experimental design was adopted to ensure unbiased estimation of main effects and potential interactions between temperature and water activity. Factorial designs are widely recommended for studying multivariable hygroscopic systems because they improve statistical efficiency and allow comprehensive evaluation of environmental influences on equilibrium moisture behavior (Montgomery, 2017). The selected temperature and water activity ranges were chosen to reflect realistic storage conditions for low-moisture food products (Labuza and Altunakar, 2007).
 
Statistical modeling
 
The combined effects of temperature (T) and water activity (a_w) on equilibrium moisture content were evaluated through statistical modeling. A multiple polynomial regression model was used to develop a predictive equation covering the full range of temperatures and water activities. This approach integrates regression analysis (for estimating functional relationships and making predictions) with factorial design principles (to assess the contributions and interactions of the factors).
       
The initial model included the linear effects of T and a_w, the quadratic term a_w² to capture curvature and the interaction term T x a_w, the model form was:
 

Mc = β₀ + β₁T + β₂𝑎_w +β₃𝑎_w² + β₄ (T x a_w) + ε

 
       
The inclusion of a quadratic a_w² term is theoretically justified by the nonlinear nature of moisture sorption phenomena, particularly in intermediate and high-water activity regions where multilayer adsorption and capillary condensation may occur (Brunauer et al., 1938; van den Berg, 1984). The interaction term (T x a_w) was initially introduced to assess potential thermo-hygroscopic coupling effects.
       
All regression analyses were performed using Minitab® 18 (Minitab LLC, State College, PA, USA). Residuals, predicted values, confidence intervals and model diagnostics were generated automatically by the software.
 
Model optimization
 
Model refinement followed standard statistical criteria, including:
• Significance of regression coefficients (p-values).
•  Goodness-of-fit indicators (R², adjusted R², predicted R²).
•  And model adequacy checks using residual analysis.
       
Although the interaction term (T x a_w) was initially considered, statistical testing revealed its non-significance (p>0.05). Moreover, multicollinearity diagnostics indicated increased variance inflation when the interaction term was retained. Model reduction was therefore performed in accordance with the principle of parsimony, which recommends selecting the simplest model that adequately describes the data while minimizing estimation variance (Kutner et al., 2005). The reduced model improved parameter stability without compromising predictive performance, as confirmed by predicted R² values:
 

Mc = β₀ + β₁T + β₂a_w + β₃ a_w²

 
This reduced model provided a better balance between simplicity, stability and predictive accuracy.
 
Validation and assumption checking
 
The validation of the model assumptions was carried out through various diagnostic tests, such as a normal probability plot, residuals vs. fitted values, residuals vs. observation order and a histogram of residuals. These tests all confirmed that the residuals were normally distributed and independent (Fig 2). The predictive capability of the model was assessed using a predicted versus experimental plot (Fig 3), while the model surface was visually represented through contour plots (Fig 4) and response surface plots (Fig 5).








       
The diagnostic analysis revealed no discernible patterns in the residual plots and a strong concordance was observed between predicted and experimental values. These findings indicate that the fundamental assumptions of linear regression normality, independence and homosce- dasticity of residuals were satisfactorily fulfilled. Consequently, the proposed model demonstrates adequate statistical validity, stability and predictive performance within the investigated experimental domain. Overall, the integration of a controlled experimental design with rigorous statistical modeling ensures both thermodynamic consistency and analytical robustness, thereby providing a reliable predictive framework for assessing the moisture-related stability of roasted and salted pistachios during storage.

Table 1 presents a summary of the experimental results obtained in the study.


       
A polynomial regression model was developed to describe the equilibrium moisture content (Mc) of pistachios, with temperature (T) and water activity (a_w) as the key determining factors. According to the data presented in (Table 2), the model is highly significant (p<0.001) and accounts for 94.52% of the total variation (R²). The adjusted R² is 93.03%, while the predicted R² is 88.16%.

​
       
The quadratic term in water activity (a_w²) exhibited the strongest effect (p<0.0001), confirming the highly nonlinear nature of the isotherm. Temperature had a moderate negative effect (p = 0.058), while the linear term in aw was marginally significant (p = 0.063), as shown in Table 3.

​
       
These findings are in agreement with previous studies conducted on agricultural and food products, where moisture sorption behavior is significantly affected by water activity and temperature interactions (Sharma et al., 2021; Patel et al., 2020). In addition, similar nonlinear trends in sorption isotherms have been reported in legumes and other biological materials, confirming the dominant effect of water activity (Kaur and Sandhu, 2022; Verma et al., 2019).

Residual diagnostics confirmed that the model met all key assumptions. The normal probability plot indicated that the residuals followed a nearly linear pattern, suggesting normality, while the histogram displayed a symmetric distribution centered around zero. The plots of residuals versus fitted values and residuals versus observation order showed no obvious pattern or heteroscedasticity (Fig 2).
       
A comparison of observed versus predicted Mc values revealed a strong agreement, confirming the predictive accuracy of the model (Fig 3).
       
The contour plot clearly illustrated the combined influence of temperature and water activity on moisture content, showing a marked increase in Mc at higher water activity values and a general decrease in Mc as the temperature increased (Al-Muhtaseb et al., 2004). The plots show that the equilibrium moisture content of the product decreases as the temperature rises at a constant water activity, a trend attributed to increased thermal motion. The figure also highlights the range of water activities where temperature has a particularly significant effect. This pattern has also been observed in pumpkin (Benseddik et al., 2018). Another study on figs found that moisture content increases as temperature decreases (Hssaini et al., 2022).
       
The 3D response surface further emphasized the dominance of the aw2 term and the curvature of the isotherm (Fig 5).
       
Using the final regression equation, the sorption isotherms of pistachio (Pistacia vera L.) kernels were predicted across the entire water activity range (0.01-0.95) for the three experimental temperatures (25^oC, 35^oC and 45^oC). The resulting curves, shown in Fig 6, display the typical Type III J-shaped profile common to high-lipid and high-soluble-solid foods. This pattern has also been noted by other researchers in a variety of food products (Karatas and Battalbey, 1991; Maskan and Karatas, 1997; Basunia and Abe, 2000; Nikoly, 2000; Saravacos et al., 1986). Moisture content increased gradually at low aw values, followed by a more rapid rise at intermediate aw and then a sharp increase above a_w ≈ 0.70. The effect of temperature was clearly evident: the isotherm curves shifted downward with increasing temperature, indicating a lower equilibrium moisture content at higher temperatures This thermodynamic behavior has been widely reported for pistachio kernels and other oil-rich seeds, where increasing temperature reduces moisture adsorption due to increased vapor pressure and reduced binding energy between water molecules and the food matrix (Mousavi et al., 2018).


       
Binding energy and increased vapor pressure of water at elevated temperatures. Overall, the modeled curves offer a smooth, continuous representation of moisture-water activity relationships and visually confirm the strong nonlinear effect captured by the aw² term in the regression model.
The final regression equation is:

Mc = 2.322 - 3.60· a_w - 0.039·T + 12.19· a_w²

 
       
Several studies have proposed alternative mathematical and intelligent modeling approaches to better describe the adsorption and desorption behavior of high-fat foods, as conventional models often fail to accurately represent sorption phenomena at high water activity levels in such products (Soleimanifard and Hamdami, 2018; Bo et al., 2017; Mokhtarian et al., 2020; Kraiem et al., 2023). More recently, advanced computational techniques, including fuzzy logic and machine learning methods, have been explored to enhance the prediction accuracy of moisture sorption isotherms in pistachio-based products (Mokhtarian et al., 2020).
       
In contrast, temperature showed a moderate yet consistent negative effect on equilibrium moisture content (Mc). This trend aligns well with previous findings for pistachios: for example, in the study by (Yazdani et al., 2006), Equilibrium moisture content decreased with increasing temperature for both pistachio powder and kernels. The negative temperature coefficient in our model reflects the increased vapor pressure and reduced adsorption affinity at higher temperatures, this results in a lower equilibrium moisture content at a given water activity. The (marginal) significance of the linear aw term in our model, along with the strong quadratic (a_w²) effect, further supports the typical sigmoidal or J-shaped sorption behavior (i.e., minimal moisture uptake at low aw, followed by accelerated uptake as aw increases) observed in high- oil/solids foods.
       
During model development, an initial regression including the interaction term (T x a_w) was tested. However, this interaction term was found to be statistically non-significant (p>0.39) and introduced high multicollinearity, without enhancing the model’s accuracy or predictive power. Upon removal of the interaction term, the model became more stable and parsimonious, with an improved predicted R². This outcome suggests that, within the studied range, the impact of temperature and water activity on moisture absorption largely operates independently of each other. This independence is consistent with some adsorption isotherm studies, where temperature shifts affect the vertical position of the isotherm curves (i.e., Mc at fixed aw) but do not noticeably alter the curvature of the aw-Mc response. For example, in a study on pistachio (Pistacia vera L.) kernels (Soleimanifard and Hamdami, 2018), temperature changes shifted the sorption curves but did not significantly alter the shape hysteresis aside.
       
Overall, the final model effectively captures the nonlinear adsorption behavior in pistachio kernels and offers a reliable predictive framework for moisture uptake under varying storage conditions. The contour and surface plots (Fig 3-4) clearly demonstrate these trends, emphasizing the sensitivity of pistachio (Pistacia vera L.) kernels to environments with high water activity, where moisture uptake accelerates rapidly. By correlating our findings with established isotherm behavior studies in pistachio systems and other high-oil foods, the model gains both empirical and mechanistic validity.

The uncoded regression equations establishing the relationship between moisture content (Mc) and hygroscopic and thermal parameters were analyzed. The resulting model, Mc = 2.322 - 3.60• a_w - 0.039•T + 12.19• a_w², demonstrates a high predictive capacity, with a correlation coefficient of 94.52%. Analysis of the coefficients reveals that water activity is the dominant factor governing moisture content, particularly due to the quadratic term a_w², which indicates a significant nonlinear influence. In contrast, temperature has a minor effect, as evidenced by its low regression coefficient (-0.039). Thus, the developed uncoded regression model serves as a valuable tool for optimizing pistachio storage conditions, allowing moisture content to be primarily estimated based on water activity, with a much smaller influence from temperature.

The authors declare that there are no conflicts of interest regarding the publication of this article. No funding or sponsor- ship influenced the design of the study, data collection, analysis, decision to publish, or preparation of the manuscript.