The number of the experiments that can be held during research is limited, because of certain causes, including economical aspect. Thus, in all fields, for the theoretical study of a certain process, the mathematical model is established, an algorithm that describes the evolution in time or after certain mutual dependencies between quantities. When establishing the mathematical model, simplifying hypotheses are adopted and approximations are made on the parameters of the analyzed process. The hypotheses adopted and the approximations made lead to incomplete mathematical descriptions that are far from reality. For these reasons, the establishment of the mathematical model (possibly) is frequently prefigured theoretically and then finalized on the basis of experimental data [1] - [6]. The analysis of the experimental data highlighted the existence of more or less accentuated nonlinear dependencies; for this reason, mathematical models should be nonlinear, the most widely used are polynomial models. Polynomial models may or may not provide explicit analytical expressions. The aim of this paper is to apply polynomial mathematical models that provide explicit analytical expressions to obtain full information for modeling the extraction processes of bioactive compounds from plant sources.

Figure 1 and Figure 2 show the results of using the cubic Hermite polynomial to establish mathematical models for hawthorn extracts, which provide the average values of parameters P1, P2, P5 and P6 depending on the concentration of ethyl

alcohol (C_a). In these examples as calculation data were adopted the average values of the parameters at each of the 5 concentrations of ethyl alcohol—20%, 40%, 50%, 60% and 80% (v/v).

Figure 1(a) shows the values of parameter P1 at two concentrations that are not found experimentally. It was found that at a concentration of 35% (v/v) it is estimated that the average value of parameter P1 is 85.1 µmol AAE/3g (point A); similarly, at 65% ethyl alcohol it is estimated that the mean value of parameter P1 is 221.1 µmol AAE/3g (point B). Figure 1(b) shows the average value of the parameter P2 at the concentration of ethyl alcohol of 25% (v/v), which is not found experimentally—44.2 µmol TE/3g (point C).

Although it operates with average values, in Figure 1 and Figure 2 are presented all the experimental values of the four parameters. Figure 1 also shows the related modeling errors.

The graphs in Figure 1 and Figure 2 showed that mathematical models based on Hermite polynomials do not provide explicit analytical expressions. Instead, these models allow the establishment, even in the calculation algorithm, of the parameter values at any concentration of ethyl alcohol, obviously in tabular form (in addition to the corresponding graph). An example in this respect is presented in Table 3 for the average values of the 6 parameters from hawthorn extract and for ethyl alcohol concentrations with a calculation step of 5%.

Figure 3 and Figure 4 show the polynomial mathematical models of the dependence between the parameter P1 and the concentration of ethyl alcohol C_a in the case of grape marc extracts.

Figure 3 and Figure 4 show that only in the case of 20^th degree polynomial (Figure 4(b)) the modeling error for the nominal model is below 2% (value imposed for reasons of precision of the mathematical model). Moreover, only in this case the uncertain models (upper and lower 95%) frame all the experimental data. Thus, in order to satisfy the first two conditions mentioned above (out of the 3 stated), in some cases polynomial models of a high degree can be obtained, sometimes difficult to use in practice. This aspect is obviously caused by the accentuated nonlinear character of the dependencies between the sizes.

For parameters P1 and P2 from aronia extracts, the 5^th degree polynomial is sufficient, as shown in Figure 5.

Figure 5(b) shows the values of the parameter P2 at the ethyl alcohol concentration of 35% (v/v), non-existent experimentally. It was found that the average (nominal) value is 30.1 µmol TE/3g, and the values in conditions of uncertainty are 24.9 µmol TE/3g and respectively 35.2 µmol TE/3g. In each of the two graphs are presented the analytical expressions of the nominal mathematical models (with index n), as well as of the upper uncertain models (index up) and lower (index l). For example, for parameter P1 in Figure 5(a) the upper uncertainty model (upper curve) has the equation:

P 1 u p = − 0.000014 C a 5 + 0.0032 C a 4 − 0.259 C a 3 + 9.62 C a 2 − 166.25 C a + 1482.58 (4)

the nominal model (middle curve):

P 1 n = − 0.000014 C a 5 + 0.0032 C a 4 − 0.259 C a 3 + 9.62 C a 2 − 166.25 C a + 1439.12 (5)

and the lower uncertainty model (lower curve):

P 1 l = − 0.000014 C a 5 + 0.0032 C a 4 − 0.259 C a 3 + 9.62 C a 2 − 166.25 C a + 1395.66 (6)

As it can be seen, the difference between the models consists in the existence of other values of the free term, explicable by the symmetry of the confidence intervals.

The graphs in Figure 6 and Figure 7 show that for the parameter P3 from the Aronia extracts a model of 5^th degree polynomial is sufficient (Figure 6(a), where the expressions of the mathematical models are also shown).

This is not the case of the parameters P4 (Figure 6(b)), P5 (Figure 7(a)) and P6 (Figure 7(b)), which require the 20^th degree polynomial (models that are difficult to use in practice).

Figure 8 shows that for the parameter P7 from the Aronia extracts a quadratic polynomial is sufficient (with the mathematical expressions in the graph), but for the parameter P8 a 20^th degree polynomial is required.

Figure 8(a) shows the expression of the nominal mathematical model:

P 7 n = − 0.00783 C a 2 + 0.434 C a + 68.51 , (7)

as well as those of uncertain models, the difference being also in the free term.

As it was found, in addition to the nominal models (of the experimental average values) were also established the uncertain models (of all the experimental data, which are located in the tires of the curves external to the nominal one). These last models were based on those previously presented in the data analysis under conditions of uncertainty.

If a high prediction is desired (the third condition of the targeted ones), then predictive mathematical models (upper and lower) must be established. An example of this is shown in Figure 9, for the case of parameter P2 from rosehip extracts. Similarly, Figure 10 shows the mathematical models for parameter P5 from rowan extracts. In both of these examples 5^th degree polynomials were used.

As can be seen from the last two graphs, they contain the 5 mathematical models, compared to the previous examples being in addition 2 predictive models (index p), upper (index up) and lower (index l). As it is observed, the analytical expressions of the 5 mathematical models differ in these cases also by the value of the free term, as it was obtained in the situations only of the nominal model and of the uncertain ones.

From the presented results that in some situations mathematical models are obtained on the whole sample (for all concentrations of ethyl alcohol) which require high degree polynomials, models more difficult to use in practice. One way to avoid this is to deduce polynomial mathematical models on certain intervals, i.e., between two concentrations of ethyl alcohol C_a. An example of this is shown in Figure 11 for parameters P1 and P2 from rosehip extracts, where cubic polynomials were used on certain intervals (3^rd degree), the mathematical expressions and their coefficients being shown in Figure 12.

As it can be seen from Figure 11(a), linear models cannot be used on intervals (as the line on the first interval) but only nonlinear ones, because the correlation coefficient is far from the unit value (ρ = −0.63); therefore, the usage of linear models on certain intervals will provide incorrect results. Also, based on the nonlinear model, in Figure 11(a) is shown the value of the parameter P1 = 0.11 µmol AAE/3g at a concentration of 55% of ethyl alcohol experimentally non-existent.

An important aspect must be emphasized here: in the analytical expressions of the mathematical models on intervals, the value of the concentration is introduced as a difference from the value of the left end of the interval, in this case 55 − 50 = 5. Thus:

P 1 ( 55 ) = 0.000122 ⋅ 5 3 − 0.0018 ⋅ 5 2 + 0.144 = 0.11   μ molAAE / 3   g (8)

From Figure 12(a), expression (1), it is found that, compared to the general relation of the mathematical model (3), this time the expression is of the form:

P 1 = a 3 C a 3 + a 2 C a 2 + a 1 C a + a 0 , (9)

for each of the 4 portions there are different values of the parameters a_m-i. For example, for the first part, with C_a = 20% - 40% (Figure 12(a)), there is a

non-lacunar mathematical model (it has all the terms of the 3^rd degree polynomial):

P 1 1 = − 0.0000019 C a 3 + 0.000396 C a 2 − 0.01356 C a + 0.229 (10)

Instead, as can be seen from the other three graphs in Figure 12, for the last three intervals there are lacunar mathematical models, at all a₁ = 0, and on the last interval and a₂ = 0.

Similarly, Figure 13 shows the analytical expressions of the mathematical models on intervals in the case of parameter P2 from rosehip extracts, which provide the average values of this parameter. As it turns out, all nominal mathematical models (with average values) are non-gaps, the coefficients having non-zero values.

Applying this algorithm, the average values of the parameters for different concentrations of ethyl alcohol are calculated with a calculation step of 5%.

Also, in this paper, the dependencies between the measured parameters were analyzed, not only between them (P1 - P8) and the ethyl alcohol concentration. The measured parameters are quantities that are not independent of each other.

An example of this is shown in Figure 14, which shows the interdependence between the concentration of ethyl alcohol, parameter P1 and parameter P3 from Aronia extracts, so a nominal mathematical model in the form of a multiple dependence P3 = f (C_a, P1).

The upper graphs in Figure 14 show simple dependencies of the type approached above, i.e., P1 = f (C_a) and P3 = f (C_a). In Figure 14(c) there is also a simple dependence, but between the two parameters, i.e., the model P3 = f (P1), whose analytical expression is plotted as a polynomial of degree 5. Finally, in

Figure 14(d) it is presented only under graphic form the mathematical model sought by the form P3 = f (C_a, P1), obviously a surface and resumed with details in Figure 15.

The graph in Figure 15, which also contains the expression of the mathematical model sought, shows the calculation area, all experimental values and their averages at different concentrations of ethyl alcohol.

Also, point A is marked with the three coordinates obtained from the expression of the mathematical model: at the ethyl alcohol concentration of 35%, the parameter P1 has the value 30.1 μmol AAE/3g and the parameter P3 value 1.8 mg ME/3g.

The existence of few experimental data with non-zero dispersions leads to the need to establish both uncertain models, which take the experimental uncertainties into account, and predictive models, which ensure the greater credibility of the results and conclusions, are drawn. Due to the existence of nonlinear dependencies between sizes, in some situations, the classical polynomial mathematical models deduced for all concentrations of ethyl alcohol can be complicated, which are so difficult to use in practice. In the case of using polynomial models on portions (between two ethyl alcohol concentrations), analytical expressions can be obtained that are not complicated, which are so easy to use in practice. For different types of plant extracts and various measured parameters, different mathematical models are obtained, which indicates the existence of diversified phenomena and the impossibility of establishing a single model for a certain type of extract. The existence of nonlinear dependencies both between the influencing factors and the measured parameters, and between the latter, leads to the need to establish non-linear mathematical models. In future research, polynomial mathematical models can be applied to describe the processes of extracting fat-soluble components (such as carotenoids) from plant sources.

This work was funded through AUF-MECC Project “Intelligent Models to Improve the Training Process” running at the Technical University of Moldova.

The authors declare no conflicts of interest regarding the publication of this paper.

Rusu, M., Ghendov-Mosanu, A. and Sturza, R. (2021) Application of Polynomial Mathematical Models for the Extraction of Bioactive Compounds from Plant Sources. Applied Mathematics, 12, 1126-1144. https://doi.org/10.4236/am.2021.1211072