Predictive Assessment of Mycological State of Bulk-Stored Barley Using B-Splines in Conjunction with Genetic Algorithms

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A predictive model combining a genetic algorithm and a B-spline curve designed to assess the mycological state of malting barley grain, which could be used as an effective tool supporting modern systems of postharvest grain preservation and storage.

Abstract

Postharvest grain preservation and storage can significantly affect the safety and nutritional value of cereal-based products. Negligence at this stage of the food processing chain can lead to mold development and mycotoxin accumulation, which pose considerable threats to the quality of harvested grain and, thus, to consumer health. Predictive models evaluating the risk associated with fungal activity constitute a promising solution for decision-making modules in advanced preservation management systems. In this study, an attempt was made to combine genetic algorithms and B-spline curves in order to develop a predictive model to assess the mycological state of malting barley grain stored at various temperatures (T = 12–30 °C) and water activity in grain (a_w = 0.78–0.96). It was found that the B-spline curves consisting of four second-order polynomials were sufficient to approximate the datasets describing fungal growth in barley ecosystems stored under steady temperature and humidity conditions. Based on the designated structures of B-spline curves, a universal parameterized model covering the entire range of tested conditions was developed. In the model, the coordinates of the control points of B-spline curves were modulated by genetic algorithms using values of storage parameters (a_w and T). A statistical assessment of model performance showed its high efficiency (R² = 0.94, MAE = 0.21, RMSE = 0.28). As the proposed model is based on easily measurable on-line storage parameters, it could be used as an effective tool supporting modern systems of postharvest grain treatment.

1. Introduction

Barley is one of the staple raw materials used to produce both human food and animal feed. In 2021, its world production reached 146-million tonnes, which accounted for 5% of global cereal production (3070 million tonnes) and ranked barley fourth following maize (39%), rice (26%), and wheat (25%) (www.faostat.org (accessed on 24 February 2023)). One of the barley usage directions is connected with its germination under controlled conditions, leading to an intermediate product: malt, which is the main component used not only in beer brewing, but also in the distilling, confectionery, fermentation, and pharmaceutical industries, among others. Barley intended for malting must exhibit a high germination capacity and energy. Therefore, this cereal grain is subject to stringent quality requirements [1].

The technological and microbiological quality of freshly harvested grain significantly depend on environmental conditions during its growth and maturation, including abiotic (temperature, water, and nutrition stress) and biotic (weeds, pests, and microorganisms) factors [2,3,4]. Excess rainfall can cause the ripening grain to be affected by diseases induced by microorganisms, e.g., fungi of the Fusarium genus (head blight or black dot disease) or diseases induced by various types of viruses (barley yellow dwarf, barley yellow mosaic), which may reduce crop yield, deteriorate grain quality, or increase grain susceptibility to spoilage during further storage [4,5,6]. Cereal grain stored in a warehouse or silo, along with the accompanying contaminants, pests, and microorganisms, constitutes a system of living organisms, referred to as the ecosystem of stored grain, which may be subject to various, also unfavorable, phenomena. Among microorganisms inhabiting harvested grain fungi, which exhibit the greatest capacity to damage grain and may contribute to economic losses in the cereal industries [7,8]. These microorganisms are equipped with a well-developed enzymatic apparatus facilitating penetration through the pericarp into the kernels. Thus, even a slight initial contamination with fungal propagules (spores, sclerotia, mycelium fragments) under conditions conducive for their development may lead to their rapid growth, resulting in heating, loss of dry matter of grain, deterioration of its nutritional and technological quality, or even complete spoilage of harvested grain mass [6]. Moreover, the presence of microscopic fungi leads to kernel discoloration and the formation of a moldy off-flavor, which has a negative effect on the quality of derived products [4,7,9]. The presence of molds also poses a health hazard to consumers, as toxic metabolites produced by these microorganisms, being extremely stable chemically and practically impossible to remove, may penetrate the human food chain through plant origin products, as well as animal origin products, contaminated with these compounds through contaminated feed [10,11]. For this reason, to protect consumer health, the Commission of the European Communities imposed more stringent quality requirements specifying maximum admissible mycotoxin contamination levels for cereal grain and their processing products, i.e., at 3 μg·kg⁻¹ for processed grain or for grain for direct human consumption, as well as 5 μg·kg⁻¹ for unprocessed cereal grain (Commission Regulation No. 466/2001 of 8 March 2001 with later amendments).

Despite the fact that the standards of postharvest grain treatment have improved substantially, the deterioration of cereal grain quality at this first stage of food processing still remains a serious problem for producers of plant-origin raw materials. According to the Food and Agriculture Organization of the United Nations, 25–33% of world cereal crops are lost each year during storage (FAO, 2018). In view of the data, proper postharvest grain preservation and storage appear to be key stages of the production and processing chain for the safety of food products derived from it. In practice, to protect crop grain against adverse changes in its bulk, grain after harvest is usually subjected to respective preservation operations consisting in the effective reduction of its moisture content (in the drying process) and then maintaining it at a safe level. One of the methods to reduce excess moisture content in plant raw materials is high-temperature drying, i.e., ventilation with heated air. This approach, within a relatively short time, protects grain against spoilage. However, on the other hand, it introduces a risk of its quality deterioration (e.g., reduction of germination capacity and energy or degradation of labile nutrients), resulting from the application of high temperature [12]. The detrimental effects of high temperatures mean that not all raw materials can be preserved by this method. In view of specific requirements for malting barley resulting from the need to retain high grain germination capacity and energy [1], near-ambient drying (in which unheated air is the drying agent [13]) seems to be better for its postharvest treatment. This preservation method ensures mild drying conditions and, thus, low energy consumption. However, on the other hand, in temperate climates (as in most countries of Central and northern Europe, as well as agricultural regions in Canada and the USA), due to the relatively low drying potential of air determined by weather conditions, this process is long term and may last from several to over a dozen days [13,14,15]. As a result, in the course of near-ambient drying for a relatively long time, the moisture content of barley grain found in the upper layer of the silo is close to its initial value, which poses a risk of mold growth in the bulk of preserved raw materials. Hence, this grain preservation method, despite its advantages, requires continuous monitoring, particularly in terms of the activity of mycobiota inhabiting the grain.

In the practice of storing raw plant materials, new, effective, and rapid methods involving tools based on near infrared spectroscopy (NIR) [16] or the e-nose technique [9,17] are being investigated to monitor the microbiological status of grain. The proposed solutions provide promising results. However, their applicability is limited to the detection of fungi that are already present in the grain mass. Meanwhile, universal, simple, and rapid solutions are more needed, enabling producers to estimate the hazard related to fungal growth in advance and, thus, undertake preemptive actions to protect the grain mass against contamination with mycoflora and the formation of mycotoxins. The answer to this demand can be provided by predictive models that would facilitate the assessment of threats related to the deterioration of plant material quality, and it might be applied as elements supporting decision-making processes in advanced management systems for postharvest grain preservation. Research on such prognostic tools combining knowledge in microbiology, mathematics, and IT for modern postharvest systems is still ongoing. Numerous studies have been conducted on the use of sigmoidal models in the form of the modified Gompertz equation [18] and the Baranyi–Roberts equation [19], reflecting the typical microbial growth curve to describe the influence of abiotic factors on the growth of individual fungi in agar media or cereal matrices under laboratory conditions [20,21,22,23]. In recent research, the modified Gompertz equation has also been successfully applied to describe the behavior of fungi in the mass of stored grain and rapeseed, depending on temperature and humidity conditions [24,25]. Nevertheless, it is worth emphasizing that the development of kinetic models based on the above equations requires the use of data that meet assumptions related to the structure of the used equation. Thus, their use is limited when microorganism development has not covered all stages of a typical growth curve. This drawback is not present in the regression based on the B-spline, which, using the so-called knots and control points, provides a better fit of the curve to empirical data compared to the classic regression. The principle of approximation using the B-spline technique consists in dividing the domain of the variable into subsets and fitting them with piecewise curves [26]. The optimal number and location of control points and knots, ensuring efficient approximation of described relationships, is determined during the B-spline optimization process. As a result, information contained in modelled data is “compressed” in the system of knots and control points shaping the B-spline curve. Thanks to its advantages, the B-spline regression technique has found applications not only in computer graphics [27], but also in other fields, including biology and life sciences [28,29,30,31,32].

Recently, methods of artificial intelligence imitating phenomena occurring in nature and operating well in modeling non-linear systems are gaining in importance in the description of microbial growth. One of them is the artificial neural network (ANN) technique being an analogy to the human nervous system, whose architectures based on structural units, i.e., artificial neurons arranged in layers, are internally linked by a dense network of connections. The signal flow and transformation in such models is carried out in accordance with a specific mathematical algorithm. ANNs are very useful tools in modeling because they are able to perform complex and accurate approximations without special assumptions [33]. For many years, investigations on the application of neural networks in predictive modeling have focused mainly on the description of growth in single bacteria species or modeling of kinetic growth parameters (e.g., maximum growth rate) of fungal strains [34,35] and the accumulation of mycotoxins associated with their development [36,37]. In recent years, researches using ANNs have also been devoted to the modeling of mold development in cereal and oilseed ecosystems treated holistically [38,39]. These studies showed a slight advantage of ANN over the use of kinetic models for the prediction of fungal growth in the stored mass of grain. Another artificial intelligence technique that has been found useful in the optimization of various non-linear problems uses genetic algorithms (GAs) [40,41]. This technique consists of searching for the optimal value/solution of the objective function by imitating biological evolutionary processes such as crossover and mutation [41,42]. In most cases, there are two representations of the analyzed problem that, by analogy with the living world, can be called genotype and phenotype [43]. A genotype is an abstract, primary-level representation (in the living world it is a sequence of purine and pyrimidine alkalis, while in the GAs, it is a set of parameters that allows for the creation of a full-fledged solution) on which crossover and mutation processes operate [43,44]. The phenotype is a set of features developed on the basis of the genotype (in the living world in the processes of gene expression by e.g., translation, transcription), whereas in genetic algorithms, it is created in the process of converting the genotype into a full-fledged solution using a set of rules [44]. The selection process may evaluate the found solutions directly (e.g., while searching for the extreme [minimum, maximum] or by means of the so-called objective function [e.g., the error between the value returned by the genetic algorithm and the expected one]). Similarly, as artificial neural networks, genetic algorithms do not require a priori knowledge to approximate the described problem [43]. In recent years, GAs have found applications in various agricultural production control systems [40,41,45,46].

The need to develop predictive tools for control and monitoring systems for postharvest grain preservation and storage, taking into account the development of mold, is still pressing. Several models have been elaborated that describe mold growth in stored grain ecosystems based on classical equations used in predictive microbiology and neural networks [24,25,38,39]. Nevertheless, the research using modern modeling techniques to predict fungal population levels in a bulk of stored grain treated holistically on the basis of storage parameters is still needed. To the best of the author’s knowledge, neither B-spline nor genetic algorithms has been applied to develop models of fungal growth in the ecosystem of stored grain. Hence, in this study, to expand the application of artificial intelligence in agricultural systems to solve practical problems, an attempt was made to combine genetic algorithms and B-spline curves in order to develop a predictive model assessing the mycological state of malting barley grain. The proposed model based on easily measurable on-line parameters, i.e., the temperature and water activity in grain, can be used as an effective tool supporting modern systems of postharvest grain preservation and storage.

2. Materials and Methods

2.1. Experimental Data Collection

When developing prognostic tools for postharvest systems, reliable data covering the range of storage conditions found in grain storage practice are indispensable. This data should be acquired from experiments conducted in ecosystems resembling the actual ones as closely as possible. In the study, the predictive model for the assessment of the mycological state of grain was developed based on data describing mold growth in the mass of grain obtained in experiments performed by Wawrzyniak [47]. The trials were conducted for the ecosystems of barley grain with an adverse initial mycological state, reflecting the level of fungal propagules in the grain characteristic of growing and harvest seasons with heavy rainfall, stored under various temperatures (T = 12–30 °C) and water availability in grain (a_w = 0.78–0.96, where a_w is water activity in grain that in the equilibrium state is equaled to relative humidity of air in inter-grain spaces, i.e., ERH = 100 × a_w, %). A total of 16 experiments were performed, each in two replicates. The arrangement of the temperature and humidity conditions in individual experiments is presented in

Table 1. Temperature and humidity conditions applied in barley grain storage experiments and the role of obtained data sets in model development.

Temperature (°C)	Water Activity	Role of Experimental Data in Model Development
12	0.80	Learning
	0.86	Validation
	0.89	Learning
	0.96	Learning
18	0.80	Learning
	0.85	Learning
	0.91	Validation
	0.95	Learning
24	0.81	Learning
	0.85	Validation
	0.91	Learning
	0.93	Learning
30	0.78	Learning
	0.80	Learning
	0.84	Validation
	0.92	Learning

. Barley samples were systematically collected during storage (frequency of sampling varied depending on storage conditions) and microbiological analyses using serial dilution and the pour plate method described in PN–ISO 21527-2:2009 and EN–ISO 6887-1:2000 were performed to determine the level of fungal colony-forming units (CFU, cfu∙g⁻¹) in grain in triplicate for each sample. During the analyses, 1 mL samples of the specified dilutions were spread in duplicate on two sterilized plates and poured with the YGC Agar, consisting of glucose, agar, yeast extract, and chloramphenicol (100 μg∙L⁻¹). The number of CFU molds was counted using a colony counter after three days of incubation at 25 °C.

2.2. Data Preprocessing

The model for the assessment of the mycological state of grain was built on the basis of experimental data, indicating the level of CFU of fungi in grain samples expressed as the decimal logarithm of CFU (Log (CFU), log(cfu∙g⁻¹). In the modeling process, an assumption was made that the level of CFU during the period of adaptation to environmental conditions (the so-called lag-phase) remained constant. Although, in fact, it was slightly lower at some points. Then, datasets obtained in individual experimental setups were split into two groups, i.e., learning and validation sets (Table 1). In the model building procedure, learning datasets were used, whereas validation datasets were not involved in building the model but were used to verify its predictive quality. For the design of the genetic algorithm, the data were normalized so that all parameters ranged from 0 to 1.

2.3. Development of the Model to Assess the Mycological State of Grain

In the study, an algorithm comprising two combined techniques, i.e., Spline Regression (SR) and Genetic Algorithms (GAs), was proposed to develop a predictive model assessing the mycological state of malting barley grain.

2.3.1. Methodology of B-Spline

The B-spline approximation methodology described in more detail earlier by Wawrzyniak [48] is based on knots and control points. If the dataset is to be approximated with a single curve, it consists of two knots. These coordinates coincide with the first and last point from the dataset and a certain number (depending on the degree of the applied polynomial) of control points. Abscissas are calculated on the basis of knots, while the coordinates are determined in the optimization process. Increasing the number of control points (and, thus, the degree of the polynomial) usually leads to a better approximation of the data. However, in practice, instead of enlarging the curve degree (above cubic) the data is usually divided into smaller subsets and approximated piecewise with independent curves, which, thanks to the overlapping of the end knot of one curve with the initial knot of the other, seamlessly join with each other, giving the so-called B-spline curve. The accuracy of the approximation increases with the rise of the number of used knots and with the rise of the number of the aforementioned number of control points. However, in practice, a compromise is sought between the number of parameters and the accuracy of the approximation so that the developed curve reflects the general trend contained in the data but not the sum of the trend and noise (disturbances). In the study, various B-spline curve structures, consisting of 1–4 component curves (and, hence, 2–5 knots and 2–7 control points, depending on the polynomial degree, i.e., 1–3) were tested in order to approximate the datasets corresponding to the development of fungi in the individual barley ecosystems. Based on the goodness of fitting of the obtained curves to the experimental data, assessed on the basis of the correlation coefficient (R), the optimal B-spline structure was selected for further research on the construction of the model describing fungal growth in barley ecosystems stored under a wide range of storage conditions. All B-splines were designed using the LSQUnivariateSpline method from the scipy.interpolate module in the Python Programming Language.

2.3.2. Methodology of Genetic Algorithms

Genetic algorithms are an optimization strategy based on observations of behaviors occurring in the animated world [49]. Just like in nature, where successive generations of organisms in the course of evolution develop those features that allow them to survive, dominate an ecological niche and even expand, so, in genetic algorithms, successive generations of solutions approach the optimal one. In both cases, data processing is parallel and takes place in a set of organisms/solutions, known as the population [44]. Shifting the population as a whole towards optimal solutions is obtained by replacing the worst organisms/solutions with new ones produced using the mechanisms of crossover (which creates organisms/solutions that are a certain combination or average of the parents’ features) and mutation (that usually introduces a small random change in the organism/solution) [44,49]. Crossover (which is the basic reproductive mechanism) provides a chance to produce organisms/solutions that inherit the best features from their parents [44]. Mutation (which, at least in the living world, from a formal point of view, is an error and usually leads to disruption of the proper functioning of the organism) allows for the creation of features that did not occur in previous generations but may prove useful for a new organism/solution [44]. Similarly, as in nature, also in GAs, organisms created by crossover and mutation are subject to selection, which regulates the probability of survival of the organism and determines the chances of its features (created by crossover and mutation) being passed on to the next generations. In general, the construction of a genetic algorithm consists in proposing: (1) A way of representing the problem (genotype); (2) Crossover and mutation operators; (3) A method of creating full-fledged solutions of the problem based on the original representation (i.e., converting genotype to phenotype); and (4) A mechanism for evaluating and selecting solutions [44].

In the study, the genetic algorithm was proposed to modulate control points of the designated B-spline using values of the storage parameters. This approach allowed for the elaboration of a model universal for all considered barley grain ecosystems. In order to develop a genetic algorithm to optimize the above-mentioned problem, the coding method (genotype) was adopted in the form of a sequence of 18 real numbers x_i (genes, which values were initially randomly assigned and then were modified using the genetic operators, i.e., crossover and mutation) acting as coefficients in equations describing coordinates of B-spline control points y_j. The values of x_i encoded, according to the method presented in

Table 2. The method of encoding the genotype of organisms used in the genetic algorithm procedure to modulate coordinates of control points (CP) of B-spline.

CP No.: j	1			2			3			4			5			6
CP value: yi	y1			y2			y3			y4			y5			y6
Factor	aw	T	aw × T	aw	T	aw × T	aw	T	aw × T	aw	T	aw × T	aw	T	aw × T	aw	T	aw × T
Gene no.: i	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
Gene value: xi	x1	x2	x3	x4	x5	x6	x13	x8	x9	x10	x11	x12	x13	x14	x15	x16	x17	x18

, were converted into y_j using the following rule:

$$y_j=x_{3j-2}·a_w+x_{3j-1}·T+x_{3j}·a_w·T,\begin{array}{cc}& j=1,\dots ,\end{array}6$$

(1)

where i = f(j) is the subsequent coefficient (gene) number from 1 to 18, j is the number of subsequent control points from 1 to 6, a_w—water activity in grain, T—storage temperature (°C), (Table 2).

While searching for the best model, organisms with the genotype coded using 18 real numbers (x_i) as mentioned above (Table 2) were subjected to the crossover and mutation operators. In the case of the first operator (crossover), the procedure started with a random selection of two organisms from the population and division (also random) of their genotype into two subsets. In the next step, genes belonging to the first and second subsets from the first and second organisms, respectively, were taken and transferred to the offspring organism (

Table 3. The crossover procedure providing the new output organisms/solution containing its genotype features (genes) of two parents (Input Organisms 1 and 2) used in the genetic algorithm for modulation of coordinates of control points (CP) in B-spline.

CP No.: j	1			2			3			4			5			6
CP value: yi	y1			y2			y3			y4			y5			y6
Factor	aw	T	aw × T	aw	T	aw × T	aw	T	aw × T	aw	T	aw × T	aw	T	aw × T	aw	T	aw × T
Gene no.: i	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
Gene value of input organism 1: 1-xi	1-x1	1-x2	1-x3	1-x4	1-x5	1-x6	1-x7	1-x8	1-x9	1-x10	1-x11	1-x12	1-x13	1-x14	1-x15	1-x16	1-x17	1-x18
Gene value of input organism 2: 2-xi	2-x1	2-x2	2-x3	2-x4	2-x5	2-x6	2-x7	2-x8	2-x9	2-x10	2-x11	2-x12	2-x13	2-x14	2-x15	2-x16	2-x17	2-x18
Gene value of output organism: 1/2-xi	1-x1	2-x2	1-x3	1-x4	2-x5	2-x6	1-x7	2-x8	1-x9	2-x10	1-x11	1-x12	2-x13	1-x14	2-x15	2-x16	1-x17	2-x18

). The resulting organism/solution has a chance to be superior to its parents by inheriting their best features.

The introduction of the newly created organism in the population was decided by the selection mechanism, which has the task to not only improve the quality of solutions, but also to maintain its high diversity. The application of the principle maintaining population diversity allows for the increase of the space of the analyzed solutions and to avoid premature convergence towards a local minimum, but its implementation required a measure of diversity in organisms to be defined. Taking it into account, in this study, the distances between individual organisms m and n (distance_mn) and the average distance in the population (distance_pop) were designated according to the following expressions:

$${distance}_{mn}=\frac{1}{k}\sum _{i=1}^{i=k}|x_{i\left(m\right)}-x_{i\left(n\right)}|$$

(2)

$${distance}_{pop}=\frac{1}{{popSize}^2}\sum _{m=1,n=1}^{\begin{array}{c}m=popSize\\ n=popSize\end{array}}{distance}_{mn}$$

(3)

where: k—number of genes (18); i—gene number; m, n—organism number; popSize—population size (number of organisms in the population); distance_mn—distances between individual organisms m and n, distance_pop—average distance in population; and x_i(m)—the i-th gene in the m-th organism, x_i(n)—the i-th gene in the n-th organism.

Another important element of the genetic algorithm comprises selection rules, which, in this study, took the following form:

• If a new organism exhibited characteristics superior to the best one in the population, then it unconditionally replaced the worst organism/solution in the population.
• If a new organism exhibited characteristics superior to the worst one in the population and meets the requirement to maintain population diversity that the smallest of the distances between the new organism and any of the organisms existing in the population was greater than the limit value, defined as 1/3 of the current average distance in the population.

In the case of the second operator (mutation), the procedure started with a random selection of one organism from the population and continued with the division (also random) of its genotype into two subsets. In the next step, the genes belonging to the first subset remained unchanged, while the genes belonging to the second subset were changed by a small, random value (

Table 4. The mutation procedure providing to the new output organisms/solution containing its genotype features (genes) of one parent (Input Organisms 1) and the randomly modified new genes (N) used in the genetic algorithm procedure for modulation of coordinates of control points (CP) in B-spline.

CP no.: j	1			2			3			4			5			6
CP value: yi	y1			y2			y3			y4			y5			y6
Factor	aw	T	aw × T	aw	T	aw × T	aw	T	aw × T	aw	T	aw × T	aw	T	aw × T	aw	T	aw × T
Gene no.: i	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
Gene value of input organism 1: 1-xi	1-x1	1-x2	1-x3	1-x4	1-x5	1-x6	1-x7	1-x8	1-x9	1-x10	1-x11	1-x12	1-x13	1-x14	1-x15	1-x16	1-x17	1-x18
Gene value of output organism: 1/N-xi	1-x1	N-x2	N-x3	N-x4	1-x5	N-x6	1-x7	N-x8	1-x9	N-x10	1-x11	1-x12	N-x13	1-x14	N-x15	N-x16	1-x17	1-x18

). The resulting organism replaced the one from which it was created, provided it was characterized by a lower approximation error.

In this study, simulations searching for the best organism/solution were performed with a population of 100 organisms. A total of 25 simulations were carried out, each consisting of 25,000 generations involving one call of both crossover and mutation operators.

2.4. Statistical Evaluation of Model Performance

The efficiency of the model to assess the mycological state of malting barley grain was evaluated using statistical indices recommended for the analysis of the predictive quality of models. The agreement between the observed data and the model responses was evaluated on the basis of the coefficient of determination (R²), root mean square error (RMSE), and mean absolute error (MAE) using the STATISTICA 13 (TIBCO Software Inc., Palo Alto, CA, USA) at the significance level of α = 0.05. The bias and accuracy factors (B_f and A_f) were used to determine the average and overall distance between observations and model responses according to the expression presented in previous studies [24,39].

3. Results and Discussion

3.1. B-Spline Curve Structure Determiantion

Previous research has shown the possibility of using B-splines in solving non-linear problems. ElAshmawy [50] proved the usefulness of two-piece spline to model the milk production and somatic cell count. Restricted cubic spline models were effectively used to determine the relation between dietary vitamin C intake and the risk of depressive symptoms [51]. The interpolation with cubic splines was also successful in determining the color of wines [52]. In the study, a parameterized B-spline curve, with control point coordinates that were modulated with the values of grain storage parameters, was used to develop the model for predicting the mycological state of barley grain. The first stage of the model construction was focused on determining the structure of the B-spline curve, which would provide the best possible approximation of the experimental data collected for a barley ecosystem stored under steady storage conditions. The elements determining the structure and performance of a B-spline that need to be defined during its design are the degree of the piecewise polynomial and the number and arrangement of knots [26,53]. In the research, to identify these parameters, B-spline curves were constructed for an exemplary dataset corresponding to one of the examined ecosystems stored at 18 °C and water activity in grain a_w = 0.85. In the course of the analysis, different degrees and the number of component curves forming the B-spline model were considered. The correlation coefficient (R), which was used to describe the goodness of fit of the B-splines to the experimental data, indicated a strong correlation between observations and estimates (R > 0.90). The simulation results presented in Figure 1 show that B-splines containing linear components, despite relatively high correlation coefficients, were the worst at approximating the analyzed data. Better results were obtained for B-splines, consisting of three and four square or cubic polynomials. It is worth emphasizing that when designing a B-spline, a compromise should be made between the quality of the data approximation and the number of knots. Overly increasing the number of knots can overfit the data and increase the variance, while overly reducing the number of knots can result in small curve flexibility and restrictive function with more bias [53]. Considering the complexity and quality of the model, it was found that the B-spline consisting of four second-order component polynomials was sufficient to approximate the analyzed dataset describing the level of fungal CFU in the examined ecosystem of stored grain (R = 0.996). Since the entire study included 16 independent storage experiments and different combinations of a_w and T were used in each of them (which resulted in different fungal growth activity), the curve determined for one storage conditions was not suitable to describe the dynamics of fungal growth in others. Previous research has shown that the combination of multilayer perceptron-based neural networks with genetic algorithms allows for building a strong mathematical tool for forecasting complex and highly non-linear systems, such as the production of medicinal mushrooms [46]. Therefore, in the study, instead of developing independent curves approximating each of the experiments separately, in the further considerations (based on the number of knots and degree of the polynomial in the piecewise curves designated for an exemplary dataset, Figure 1), an attempt was made to describe the entire dataset using one parameterized B-spline curve, for which the coordinates of the control points were modulated by genetic algorithm on the basis of the values of the storage parameters (a_w and T). The control points determined in this way made it possible to design a B-spline curve that retains all its features favorably from the modeling point of view, and which course can be adjusted to the fungal growth curve depending on the storage conditions.

3.2. Genetic Algorithms Design

Genetic algorithms (GAs), which are excellent optimization tools operating analogously to natural evolution processes, have found a wide application in various types of spectral analyzes to samples selection [54] and in many combinational optimization problems [41,42,46]. In this study, a genetic algorithm was used in the process of searching for optimal coordinates of control points shaping the B-spline developed to predict the mold population levels in examined barley ecosystems. As this technique uses a multi-point search procedure that focuses on wide areas of the solution space, the application of GA allows for an opportunity to quickly and efficiently find the suboptimal value/solution, even from a very large search space [42]. As mentioned in the methodology, a population of 100 organisms was considered during the simulation. In order to find the optimal organism/solution, 25 simulations were performed (each consisting of 25,000 generations, with one crossover and mutation operator invoked within each of them). The best organism in the population generated within the single simulation was the one, which was characterized by the lowest value of the average of learning and validation errors. The results obtained in subsequent simulations, consisting of the values of learning, validation errors, and the average values of learning and validation errors for the best organism in the population and learning errors for the worst organism in the population, are presented in

Table 5. Values of errors computed during optimization procedure using a genetic algorithm (25 simulations, S): learning (L) and validation (V) errors for the best organism (B), average values of learning and validation errors (A(L + V)/2) for the best organism and learning errors for the worst organism (W) in the population.

S no.	L_B	V_B	A(L_B + V_B)/2	L_W
1	0.0773	0.0717	0.0745	0.0803
2	0.0774	0.0760	0.0767	0.0808
3	0.0778	0.0733	0.0756	0.0816
4	0.0778	0.0740	0.0759	0.0865
5	0.0781	0.0762	0.0772	0.0817
6	0.0782	0.0718	0.0750	0.0835
7	0.0783	0.0790	0.0786	0.0840
8	0.0783	0.0770	0.0777	0.0826
9	0.0784	0.0823	0.0804	0.0820
10	0.0784	0.0776	0.0780	0.0832
11	0.0786	0.0780	0.0783	0.0828
12	0.0786	0.0778	0.0782	0.0847
13	0.0789	0.0778	0.0784	0.0825
14	0.0789	0.0806	0.0797	0.0866
15	0.0794	0.0771	0.0783	0.0850
16	0.0795	0.0764	0.0779	0.0917
17	0.0798	0.0731	0.0764	0.0855
18	0.0798	0.0966	0.0882	0.0858
19	0.0801	0.0819	0.0810	0.0830
20	0.0803	0.0824	0.0813	0.0840
21	0.0804	0.0757	0.0781	0.0884
22	0.0805	0.0709	0.0757	0.0857
23	0.0809	0.0699	0.0754	0.0922
24	0.0813	0.0730	0.0771	0.0862
25	0.0838	0.0785	0.0811	0.0912

.

The presented data show that the results obtained in each of the simulations are comparable (e.g., the error for the best organism from the learning dataset ranged from 0.077 to 0.083, with the mean and median equal to 0.079). From among the best organisms registered in 25 simulations, the one with the lowest average value of learning and validation errors was selected for use in the model assessing the mycological state of malting barley grain. The course of changes in the value of the learning error for the best and the worst organism and the validation error for the best organism in the population, for a simulation, in which the best results were achieved, is shown in Figure 2. As can be seen, in the initial phase of the simulation process, the errors decrease quite rapidly. However, after exceeding the limit of 10,000 iterations (generations), the changes are much slower.

The decrease in learning rate is also confirmed by the data shown in Figure 3 presenting the number of successes for the genetic operators, where after exceeding 10,000 iterations, only a few crossover and mutation operators ended in entering the organism into the population.

It is worth noting that despite the introduction of the requirement to enforce a minimum distance between the new organism and organisms forming the population (the assumed threshold for entry into the population was lower than the average distance in the population and constituted 1/3 of its value), the average value of the distance in the population decreased during the simulation process (Figure 4). However, this process was slightly slower than in the absence of the aforementioned requirement. Replacing the worst solutions in the population with new ones not only decreased the average distance between organisms in the population but also moved the population as a whole to this part of the solution space, where suboptimal results were located.

The proposed approach of gradually narrowing the space of the analyzed solutions seems to meet the requirements of the selection process. On the one hand, this should move the population to the part of the solution space where the best of them are located. On the other hand, it should protect the population against premature convergence towards a local minimum. The obtained results confirm that the use of GAs allow to overcome the shortcomings of the general iterative methods that often are likely to fall into the “local minimum trap” [41]. A good optimization effect of the applied genetic algorithm is also evidenced by the similar nature of changes in the learning and validation error (Figure 2), which proves that the model performed well on new data (not involved in the model building process) and is characterized by a good ability to generalize.

3.3. Evaluation of Model Performance

Predicting tools dedicated to supporting systems of postharvest grain preservation and storage are highly promising and essential to maintain the quality of this raw material and nutritional safety of cereal-based products. Previous research has shown that hybrid models using genetic algorithms have been successful in optimizing and modeling complex non-linear natural growth and developmental processes [46,55,56,57]. This study is a new modeling approach and focuses on predicting the population level of fungi that contaminate cereal ecosystems using a mathematical model, which is a hybrid of the B-spline technique and genetic algorithms. The use of both of these techniques in research facilitates the elaboration of a model in which the coordinates of the control point shaping the B-spline were related to the storage conditions of the considered malting barley ecosystems. The computerized values of learning and validation errors obtained in the simulations indicated good prediction quality of the elaborated model. Nevertheless, for a more detailed assessment of the model efficiency, the statistical indicators recommended for predictive model evaluation have also been calculated. The comparison of the model responses with the observations presented in Figure 5 proves a high correspondence of the analyzed data. The RMSE and MAE values also indicate a high agreement between model predictions and experimental fungal CFU levels in barley ecosystems (

Table 6. Statistical indicators used to evaluate the performance of the developed model to assess the mycological state of malting barley grain with a_w = 0.78–0.96 stored at T = 12–30 °C.

Statistical Indices	DATASET
	Learning	Validation	Full
Number of experimental points (N)	201	63	264
Coefficient determination (R2)	0.944	0.916	0.938
Mean absolute error (MAE)	0.208	0.210	0.209
Root mean square error (RMSE)	0.278	0.247	0.276
Bias factor (Bf)	0.996	1.009	0.999
Mean relative percentage error (MRPE), (%)	−0.162	−1.058	−0.048
Accuracy factor (Af)	1.041	1.040	1.041
Mean absolute percentage error (MAPE), (%)	4.000	4.000	4.000

).

The high efficiency of the developed model was also confirmed by the values of bias indexes (B_f and MRPE), designating that the negative and positive distinctions of the model are almost equal. A negative MRPE value suggests that the model slightly tended to overestimate the level of mold in the examined grain. However, in the case of the growth of microorganisms, this can be considered as an increase in the safety of the model in its practical application. A slight overestimation of the model (which is a model of quality degradation) is not limitative [58] and means that its outputs are slightly higher than the actual levels of grain contamination with fungi, which can allow to react in time and avoid risky situations. The accuracy factor and mean absolute percentage error values presented in Table 6 are convergent and indicate that the model prediction error is about 4%. The results are comparable with those obtained in the case of models based on artificial neural networks (ANN) and the kinetic equation in the form of the modified Gompertz equation that were developed for the same data describing the level of fungal infection in barley ecosystems [24,39]. It is worth emphasizing that although the modified Gompertz equation facilitates biological interpretation of its parameters, it requires the fulfillment of certain assumptions related to the data structure. The advantage of the elaborated model based on the B-spline and genetic algorithms over the one based on the kinetic equation is that, similar to ANNs, it does not need to meet any initial assumptions. In turn, comparing the model proposed in this study with the one developed using ANN showed that the model combining B-splines with the genetic algorithm provided slightly worse results. However, on the other hand, it was characterized by a smaller number of control parameters (18 in the case of the newly proposed algorithm versus 26 (15 [3 × 5] weights between input and hidden layers, plus five [5 × 1] weights between hidden and output layers, plus six biases [one for each neuron in the hidden and the output layer] in the case of ANNs). The evaluation of the developed hybrid model showed its high usefulness for assessing the mycological state of malting barley ecosystems stored in a wide range of humidity and temperature conditions that may occur in agricultural practice. As the model describes the dynamics of fungal development in the grain ecosystem, it may be useful for assessing the allowable time of its safe storage and selection of an appropriate method of its preservation based on the temperature and humidity conditions prevailing in the grain mass. Moreover, as the proposed modeling approach is universal in character, it can be used in other approximation areas.

4. Conclusions

The increased consumer awareness and growing demand for food with high health-promoting value force cereal producers to meet more stringent requirements. One of the main threats to the technological quality of grain and the health of its consumers is fungal growth, which, in conditions favorable for its growth, can lead to its contamination with toxic metabolites: mycotoxins, even within a few days. Considering the above aspects, it appears that the best approach to ensure consumer protection and, thus, avoid toxin accumulation is the sustainable management of postharvest processes to prevent mold growth. The development of digital technologies gives the opportunity to create new predictive tools for the food industry. Artificial intelligence has recently gained a lot of interest as it facilitates the construction of models that can be incorporated into existing solutions, extending their applicability. In this study, an attempt was made to elaborate the hybrid model for predicting the mycological state of barley grain using a parameterized B-spline curve, in which the genetic algorithm was responsible for the modulation of coordinates of the control points based on grain storage parameters, i.e., temperature and water availability. The high efficiency of the elaborated model describing microbial growth showed the usefulness of the applied B-spline modeling technique and genetic algorithms for the development of exploratory tools that can find practical applications in supporting postharvest grain decision-making processes. The utilitarian significance of the developed model is enhanced by the fact that it is based on easily measurable on-line abiotic factors determining the intensity of changes occurring in the ecosystem of stored grain and having the greatest effect on fungal growth and mycotoxin production.