Before frying, the processing of the fish consisted of cleaning, trimming and cutting. The fish were cleaned with clean water to remove impurities. Trimming was then carried out and the head of each fish was cut and separated from the rest, then the viscera was removed. The fish thus treated was further cleaned and then cut. The tuna pieces were then sprinkled with cassava flour and introduced into the oil (refined palm oil from the ‘‘Dinor’’ brand) heated to the desired temperature using an electric fryer. The fryer used in this study is Moulinex brand (Serial No. AF102030, France) with fixed tank with non-stick coating, allowing adjustment and control of the frying temperature.

After frying, the content of newly formed compounds in the oils (PAH, acrylamide) was determined according to the standardized method NF EN ISO 8420 [9] using Testo 270 (No. 20388792, Germany).

The peroxide index of the oils was determined according to the method of the International Union of Pure and Applied Chemistry 2.501 [10] .

The acid index of the oils was determined according to the method of the International Union of Pure and Applied Chemistry (IUPAC) 2.201.

In order to preserve the quality of fried tuna, modeling following the box-Behken plan was applied to obtain optimal values of the independent variables (temperature, number of reuses, frying time).

The choice of factors for this study is based on the results of the diagnostic study of frying practices [11] but also on the results of certain studies on frying [12] - [14] . The factors examined are:

The different factors were provided with the same number of modalities (3 levels) in the application of the Box-Behnken plan methodology. Table 1 presents the factors studied and the variation for each factor, namely the low (−1), medium (0) and high (1) value, represented respectively by their coding (x₁, x₂, x₃).

The answers chosen in this study are the parameters reflecting the degradation of oil during frying. These responses are the content of newly formed compounds (%), the peroxide index (meqO₂/kg) and the acid index (mg of KOH/g) after the frying tests of each combination given by the plan of experience.

The number of experiments (N) necessary for the development of the Box-Behnken model is defined as follows: N = 2k (k − 1) + C₀, (k is the number of factors and C₀ is the number of points at the center of the domain experimental) [15] . Thus, the experimentation matrix is given by Table 2 and is composed of 15 experiments.

A second-degree polynomial function (Equation (1)) was postulated in order to predict the optimal point for which the oil still has the quality recommended for direct consumption and to be used as an ingredient for frying a food.

$y=a_0+a_1{x}_1+a_2{x}_2+a_3{x}_3+a_{11}{x}_1^2+a_{22}{x}_2^2+a_{33}{x}_3^2+a_{12}{x}_1{x}_2+a_{13}{x}_1{x}_3+a_{23}{x}_2{x}_3+e$ (1)

with y = response or dependent variable, a₀ = model constant, a_i = linear coefficients, a_ij = interaction coefficients and a_ii = quadratic model coefficients.

(X₁), (X₂) et (X₃) correspond respectively to the factors frying temperature, number of oil reuse, tuna frying time, C = central points.

A second-degree polynomial function (Equation (1)) was postulated in order to predict the

1) Modeling and optimization of the frying process

The industrial statistical module & six sigma (experiment plan) of the Statistica 10 software was used to establish the Box-Behnken plan matrix and carry out the various statistical modeling and optimization treatments.

2) Analysis of variance

Assessment of the overall quality of the developed model was carried out using an ANOVA and the Fisher-Snedecor test. If the Fobs value for regression (Equation (2)) is greater than the corresponding F_crit value (Equation (4)), and the Fobs value for lack of fit (Equation (3)) is less than the F_crit value (Equation (4)) corresponding we conclude that the mathematical model has a quality of adjustment (Montgomery, 2013).

$F_{\text{obs}}=\frac{MC{E}_{\text{regression}}}{MC{E}_{\text{residual}}}$(2)

$F_{\text{obs}}=\frac{MC{E}_{\text{lack of adjustment}}}{MC{E}_{\text{pure error}}}$(3)

$F_{\text{crit}}=F\left(0,05;dd{l}_{\text{numerator}};dd{l}_{\text{denominator}}\right)$(4)

With, F_obs = calculated F-ratio value and F_crit = F-ratio value read in the Ficher table; MCE_regression = mean of the squares of the differences associated with the regression; MCE_Residual = average of the squares of the deviations associated with the residuals MCE_{lack of adjustment} = average of the squares of the deviations associated with the lack of adjustment; MCE_{pure error} = average of the squares of the deviations associated with the pure error.

3) Response optimization method

Response optimization was calculated using the desirability function approach. The calculation software then gives one or more sets of optimal conditions accompanied by an indicator, called the desirability coefficient (d_i) for each of the observed responses (Equation (5)). We then determine, where applicable, the total desirability (D), defined as the geometric mean of the individual desirabilities (d_i) (Equation (6)) [16] .

$d_i=\frac{1}{M}{\displaystyle \sum w_i{\left(\frac{y_i-T_i}{T_i-L_i}\right)}^2}$(5)

$D={\left(d_1\times d_2\times \cdots \times d_k\right)}^{\text{1}/\text{4}}$(6)

With, M = number of responses, y_i = responses, W_i = weight granted to each response (we will take it equal to 1), T_i = response target value and L_i = acceptable limit value; D = total desirability; et d_i = individual desirability ( $i=1,2,\cdots ,K$).

During the performance of the Box-Behnken design tests for three factors, three responses (Y₁: % newly formed compounds; Y₂: peroxide index, Y₃: acid index) were measured for this study. The values of these responses as well as the responses predicted by the plan (BB) are recorded in Table 3 .

1) Effect of independent variables on the content of neoformed compounds

The iso-response curves ( Figure 1(A) ) show that the content of newly formed compounds increases as the number of times the oil is reused and the frying temperature increases. The content of 25% of newly formed compounds was obtained from temperatures above 160˚C for 4 reuses of the oil ( Figure 1(A) ).

Analysis of the combined effect of temperature (X₁) and frying time (X₃) shows that contents of less than 25% of newly formed compounds are obtained for temperatures between 140˚C and 160˚C and frying times. 6 minutes and 8 minutes. Levels above 25% are achieved by applying temperatures above 160˚C and frying times above 10 minutes ( Figure 1(B) ).

As for the combined effect of the frying time (X₃) and the number of reuse (X₂) with the temperature factor (X₁), the content of newly formed compounds increases with the increase in the number of reuse and the frying time.

2) Mathematical model retained for the response in neoformed compounds

All coefficients of the linear term (X₁, X₂, X₃) are significant (p ≤ 0.05), which indicates that the three factors have a significant influence on the evolution of the neoformed compounds during frying. But the factor X₂ presents the lowest P-value (p ≤ 0.001), which highlights the great impact and influence that this factor has on the production of neoformed compounds in the tuna frying process. Also, all coefficients of the quadratic term ( ${\text{X}}_3^2$, ${\text{X}}_1^2$, ${\text{X}}_2^2$) are significant (p ≤ 0.05). Only two coefficients of the interaction term (X₁X₃ et X₂X₃) are not significant (p ≥ 0.05) ( Table 4 ).

The mathematical model retained is given by the following Equation (7):

$\begin{array}{l}{\text{Y}}_1{}_{\left(\text{neoformed compounds},\%\right)}=15.16+5.56250{\text{X}}_1+13.56250{\text{X}}_2+2.75000{\text{X}}_3\\ +3.41667{\text{X}}_1^2+2.91667{\text{X}}_2^2+5.04167{\text{X}}_3^2+2.75000{\text{X}}_{\text{1}}{\text{X}}_2\end{array}$(7)

3) Assessment of the fit quality of the postulated model

The higher Fobs value for the model compared to the F_crit value from the Fisher table and the Fobs for poor fit compared to the F_crit value indicates that the model has a good fit. Also, the regression line of measured responses versus predicted values shows a strong correlation between them. The figure indicates that the points are well distributed around the regression line. The model can therefore be judged to be of sufficient quality, since it explains 96.87% of the measured variations in the response in newly formed compounds with an R² (adjusted) = 0.91259.

ES = standard error, t-exp = calculated student value, p-value = probability associated with the value of t, a0 = constant of the equation, (a1, a2, a3) = linear coefficients, (a11, a22, a33) = quadratic coefficients, (a12, a13, a23) = interaction coefficients, (ns) = not significant (p > 0.05), (**) = highly significant (p ≤ 0.001), (*) = weakly significant (0.001 < p ≤ 0.05).

1) Effect of independent variables on peroxide index

The iso-response curves ( Figure 2(A) ) indicate that the reuse of the oil for frying at high temperature results in an increase in the peroxide index of the oil. A peroxide index of 10 meqO₂/kg was obtained beyond the 7th reuse for temperatures of 140˚C and 160˚C ( Figure 2(A) ).

The combined effect of temperature (X₁) and frying time (X₃) shows that low peroxide index values (4 and 6 meqO₂/Kg) are obtained for temperatures between 140˚C and 170˚C and frying times of 5 minutes and 7.5 minutes. While values of 10 meqO₂/Kg are achieved by applying temperatures above 170˚C and frying times above 8.5 minutes ( Figure 2(B) ).

In Figure 2(C) , there is an increase in the peroxide index with oil reuse and frying time. The peroxide value of 10 meqO₂/kg is reached at the 8th reuse of the oil for frying times of 10 minutes when the temperature is set at 160˚C.

2) Mathematical model for the response “peroxide index”

All coefficients of the linear term (X₁, X₂, X₃) are significant (p ≤ 0.05), with a p-value ≤ 0.05 for the factor X₂ (number of reuses). Thus, only two coefficients of the quadratic term ( ${\text{X}}_1^2$, ${\text{X}}_3^2$) and one coefficient of the interaction term (X₁X₂) are significant (p ≤ 0.05) ( Table 5 ).

The mathematical model chosen is given by the following Equation (8):

$\begin{array}{l}{\text{Y}}_{2\left({\text{Peroxide index in meqO}}_2/\text{kg}\right)}=4.541667+2.234375{\text{X}}_1+5.484375{\text{X}}_2+0.750000{\text{X}}_3\\ +2.494792{\text{X}}_1^2+1.463542{\text{X}}_3^2+1.093750{\text{X}}_1{\text{X}}_2\end{array}$ (8)

With Y₂ being the peroxide index response and X₁, X₂ and X₃ being the independent variables; frying temperature, number of reuses and frying time.

3) Assessment of the fit quality of the postulated model

Moreover, analysis of the graph of the curve of the peroxide index values measured as a function of the predicted values shows that the curve has the shape of a straight line, which also explains a close agreement between the experimental results and the values predicted by the polynomial model. The predicted and measured responses are therefore well correlated, with coefficients of determination (R² = 0.99287); (R² (adjusted) = 0.98002) greater than 0 and equal to 1.

1) Effect of independent variables on acid value

The iso-response curves show that the reuse of the oil and the temperature essentially have an action on the evolution of the acid value of the oil ( Figure 3(A) ). The acid value of 4 mgKOH/g was obtained for oil reuse of more than three from 150˚C ( Figure 3(A) ).

The effect of temperature (X₁) and frying time (X₃) on the acid number is also illustrated in Figure 3(B) with the number of reuses set at its zero level (5). The increase in the acid number is due to the combined effect of temperatures between 140˚C and 180˚C and frying times of 5 to 10 minutes.

Figure 3(C) gives the curves as a function of the frying time (X₃) and of the number of re-uses (X₂) with the temperature factor (X₁) fixed at its zero level. The

SE = standard error, t-exp = calculated student value, p-value = probability associated with the value of t, a0 = constant of the equation, (a1, a2, a3) = linear coefficients, (a11, a22, a33) = quadratic coefficients, (a12, a13, a23) = interaction coefficients, (ns) = not significant (p > 0.05), (**) = highly significant (p ≤ 0.001), (*) = weakly significant (0.001 < p ≤ 0.05).

acid number increases with oil reuse and frying time. Its highest value is reached at the 9th reuse of the oil for frying times of 10 minutes.

2) Mathematical model for the acid index response

The results of the estimation and significance of the regression coefficients of the postulated Box-Behnken second-degree model, representing the relationship between the change in acid index and the three independent variables studied, indicate that the model is significant (p ≤ 0.05) ( Table 6 ). Thus, all the coefficients of the linear term (X₁, X₂, X₃) are significant (p ≤ 0.05), with a strong significance for the factor X₂ (number of re-uses) which has a p-value = 0.001. Also, the terms of the coefficient of the quadratic term ( ${\text{X}}_3^2$, ${\text{X}}_1^2$, ${\text{X}}_2^2$) are significant and in the interaction term only the combined action of temperature and number of reuse (X₁X₂) is significant (p ≤ 0.05).

The mathematical model used is given by the following Equation (9):

$\begin{array}{l}{\text{Y}}_3{}_{\left(\text{Acid index in mgKOH}/\text{g}\right)}=3.428333+1.651875{\text{X}}_1+2.620625{\text{X}}_2+0.566250{\text{X}}_3\\ +1.130833{\text{X}}_1^2-0.914167{\text{X}}_2^2+0.994583{\text{X}}_3^2+0.780000{\text{X}}_1{\text{X}}_2\end{array}$(9)

where Y₃ is the acid index response and X₁, X₂ and X₃ are independent variables; frying temperature, re-use number and frying time.

3) Assessment of the fit quality of the postulated model

The model is significant (p = 0.001) with a non-significant lack of adjustment (p = 0.05), according to ANOVA based on the quality of the Box-Behnken model of the relationship between the three selected independent variables and the acid index. Finally, the analysis of the curve of the acid index values measured as a function of the predicted values shows that the curve also has the shape of a straight line, which explains a close agreement between the experimental results and the values predicted by the polynomial model. The two are well correlated, with coefficients of determination (R² = 0.98127; R² (adjusted) = 0.94755) greater than 0 and close to 1, which confirms the quality of the response modeling.

SE = standard error, t-exp = calculated student value, p-value= probability associated with the value of t, a0 = constant of the equation, (a1, a2, a3) = linear coefficients, (a11, a22, a33) = quadratic coefficients, (a12, a13, a23) = interaction coefficients, (ns) = not significant (p > 0.05), (**) = highly significant (p ≤ 0.001), (*) = weakly significant (0.001 < p ≤ 0.05).

1) Determination of optimal conditions

The values of the response optimizing factors (Y₁: neo-formed compounds, Y₂: peroxide number, Y₃: acid number) at the limit of the recommended values required for a consumable frying oil were determined using the desirability function of the STATISTICA 10 software ( Figure 4 ).

Examination of all the graphs allowed the selection of the coordinates of the optimal point of the three factors at their zero levels, i.e. (Temperature: 160˚C, Number of re-uses: 5 and frying time: 7.5 minutes).

2) Checking optimal conditions

Under these optimum conditions, mean values of 15.5% ± 0.33% of neoformed compounds, 4.43 ± 0.11 meqO₂/Kg in peroxide index, and 3.51 ± 0.01 mgKOH/g in acid index were obtained. While the model predicted a neoformed compound content of 15.17%, a peroxide value of 4.54 Under these optimum conditions, mean values of 15.5% ± 0.33% of neoformed compounds, 4.43 ± 0.11 meqO₂/Kg in peroxide index, and 3.51 ± 0.01 mgKOH/g in acid index were obtained, while the model predicted a neoformed compound content of 15.17%, a peroxide index of 4.54 meqO₂/Kg and an acid index of 3.43 mgKOH/g ( Table 7 ).