Plant stem tissue modeling and parameter identification using metaheuristic optimization algorithms

Bio-impedance non-invasive measurement techniques usage is rapidly increasing in the agriculture industry. These measured impedance variations reflect tacit biochemical and biophysical changes of living and non-living tissues. Bio-impedance circuit modeling is an effective solution used in biology and medicine to fit the measured impedance. This paper proposes two new fractional-order bio-impedance plant stem models. These new models are compared with three commonly used bio-impedance fractional-order circuit models in plant modeling (Cole, Double Cole, and Fractional-order Double-shell). The two proposed models represent the characterization of the biological cellular morphology of the plant stem. Experiments are conducted on two samples of three different medical plant species from the family Lamiaceae, and each sample is measured at two inter-electrode spacing distances. Bio-impedance measurements are done using an electrochemical station (SP150) in the range of 100 Hz to 100 kHz. All employed models are compared by fitting the measured data to verify the efficiency of the proposed models in modeling the plant stem tissue. The proposed models give the best results in all inter-electrode spacing distances. Four different metaheuristic optimization algorithms are used in the fitting process to extract all models parameter and find the best optimization algorithm in the bio-impedance problems.

www.nature.com/scientificreports/ the intelligence-gathering behavior of water cycle 50 , pollination process in the plant 49 , chicken behavior in the swarm 51 , red fox searching for food, hunting and escaping from hunters 52 , black widow spider mirage 53 , Harris Hawks during chasing of the prey 54 , elephants herd and the distance between them 55 , hunting process of the grey wolf 56 , and many others. They are used to overcome the defecates and difficulties that face the traditional optimization methods 49 . In 33 , flower pollination algorithm (FPA) and moth flame optimization (MFO) were used to extract the Cole-impedance model parameters and compared with the traditional nonlinear least square (NLS). FPA and MFO showed their superiority over the NLS method in fitting the measured data and accuracy of the extracted parameters. Also, the FPA achieved the best accuracy and consistency over the other employed optimization (NLS,MFO), while the NLS technique was the fastest. In 49 , six different metaheuristic optimizations were used to extract the Cole-impedance model parameters using two different datasets magnitude only impedance measurements and complex impedance measurements. It was found that Cuckoo search optimization (CS) and FPA algorithms had a better fitting for the experimental datasets, less error and higher consistency than the other used algorithms. FPA, CS, and MFO algorithms were used in 45 to extract the Cole-impedance model parameters using an alternative way to measure the bio-impedance (differentiator circuit). It was concluded that CS and FPA algorithms had a quite similar performance, where CS converges faster, and FPA takes less run time. FPA showed a reasonable parameter extraction over CS and MFO. Optimization algorithms are recognized as a soft computing method used to solve complex problems. Soft computing is concerned with approximate models and controlling complex systems, as it is tolerant to imprecision, uncertainty and approximations. Soft computing is a combination of optimization algorithms, in addition to artificial neural networks and machine learning algorithms that are used for decision-making 57 , identification 58 , and predictions support 59 .
In this paper, two new Fractional-order electrical impedance models are proposed for plant stem representation. The stem impedance is measured using SP150 for two samples of three medical plants (Marjoram, Salvia officinalis L., Lavandula) from Lamiaceae plant family. The measured impedance data are fitted on three commonly used bio-impedance models with plants (Cole, double Cole and Fractional-Order double-shell), and compared with the two proposed models. Then the models' parameters are extracted using four metaheuristic optimization algorithms [FPA, CS, WCA and Chicken swarm optimization (CSO)]. The Nyquist plot is plotted for the measured and the fitted data for all models. The error between the measured and fitted data is calculated to find the best model and the best optimization algorithm.
This paper is organized as follows: Section "Stem modeling" briefly describes the plant stem anatomy and the role of each stem layer, it also shows the impedance model circuit's analysis and their representation. Section "Problem definition" illustrates the problem formulation. Section "Experimental results and discussion" provides the experimental results and discussion. Finally, Section "Conclusion" concludes the paper.

Stem modeling
Stem tissue structure. Plant Stem plays a vital role in the growth and protection of the plant, providing support to the plant weight. It bears the flowers and leaves of the plant and acts as a transportation channel for water, nutrients and food through all the plant parts. The green stems participate in the Plant's Photosynthesis process 60 . Monitoring the plant stem helps to investigate the plant's condition, such as transpiration rate (water flow) and nutrient concentration. It could also act as an indication to the soil state 37 . For medical plants, some beneficial compounds are extracted from the plant stem, such as ethanol that has antioxidant and anti-gout activity; it also contributes to the production of some essential oils 61 . The stem (see Fig. 1) includes multiple layers that depend on the structure of the plant and its growing conditions. It mainly consists of Vascular, ground and Epidermis systems 60 . The Vascular system is composed of Xylem and Phloem as Complex cells. The Xylem is responsible for transferring water and nutrients from the roots along the whole stem and into the leaves. It is a one-directional tube that consists of smaller tubes connected through a gate. The Phloem is a bidirectional transportation system that transports food and organic materials from the green parts to the rest of the plant. The Phloem and Xylem are grouped in vertical strands called vascular bundles and are separated by a layer of cells named cambium 37,60 .
The epidermis in Fig. 1 is the outer layer that covers the stem with a rigid structure and waxy appearance in some plant species. It protects the stem against injury, infection and water loss 60 . It also acts as a controller of the gas, water and nutrients exchange with the surrounding environment. The epidermis evolved various features, such as some specific cell types and guard cells, to adjust to its various functions. The cell's shapes and functions are developed according to their growing circumstances 62 .
The Plant ground parts are responsible for the stem support, where it consists of the Pith and the cortex that is located between the vascular bundle and the epidermis. The tissue cells of the cortex may include essential oils, tannins and stored carbohydrates. The Pith is at the centre of the stem with a soft spongy structure. It contributes to the storage of nutrients and minerals. For some plants, the stem could harden and then decomposes to produce a hollow shaped stem 60 .
Electrical modeling. General bio-impedance models. The biological cell cannot be dealt with as a homogeneous medium, consisting of various complex elements 35 . When current migrates through a cell, it is attenuated by existent water electrolytes, and intracellular and extracellular components. The Cole-impedance model shown in Fig. 2a proved to produce a good fit to experimental impedance data when applied to different tissues. It was initially proposed as a general model. Then it was applied more specifically to describe plant status. The Cole-impedance model is represented as follows: where R o represents the resistance at low frequency, R ∞ represents the high-frequency resistance and α represents the Constant phase element(CPE) order. The Cole-impedance model is used to fit the measured data of various types of tissues such as shoots and leaves tissues in 63 .
It was also used to study the effect of freezing-thawing on eggplant, maturity measurements of fruits and vegetables 64 . In 65 , the ripening of fruits was investigated using various models. In 17 , the Cole-impedance model was used to study the photosynthetic activity in plants during illumination and darkness. Although this model gave good results for most of the experimented tissues, it did not provide an explanation for the operating mechanism in the cell.
To provide a better representation for cell components in complex materials, the Cole-impedance model was expanded as shown in Fig. 2b into double Dispersion Cole impedance model where its impedance representation is as follows: The double dispersion Cole impedance model was used as a representation of the plant stem in 37 . It was used to fit measurement data of various fruits and vegetables such as banana, cucumber and oranges in 35 , where different models were compared, and the double Cole impedance model provided the best results. The double Cole impedance model could be used as an indicator of frost hardening in shots of Scots pine 64 .
To accurately describe the plant tissue cell's components, the Double-shell model was developed to provide a representation of the cell Vacuole. The fractional-order Double-shell model shown in Fig. 2c was firstly introduced in 35 and its impedance is described as follows: where R 1 represents the extracellular resistance, R 2 represents the intracellular resistance, R 3 and C β represent the Vacuole resistance and capacitance respectively, and C α represents the plasma membrane capacitance. The double-shell model is used in studying different plant condition such as ripening, heating and Freezing, but proved to be most efficient in plants ripening 17 . It was also used as a representation for the plant stem structure 37 .
Proposed bio-impedance models. The proposed electrical impedance model in Fig. 2d characterizes the plant stem. The Epidermis (see Fig. 3) is a hard protective layer; in most cases, it consists of a single layer of cells represented by an electrical resistor R o . The Xylem, Phloem, and the Cambium (Bundle) are each represented by a resistor and capacitor in series as they have a tube-like structure. Also, the Cortex has the exact representation.  www.nature.com/scientificreports/ While the Spongy pith is represented by a capacitor C . The electrical impedance of the stem model is described as following: The proposed stem model is simplified as in Fig. 2e by representing the vascular bundle by a single branch of series resistor and capacitor. Its electrical impedance is described as follows: where R o represents the Epidermis,R 1 and C α represents the Cortex resistance and capacitance respectively, R 2 represents the resistance, and C β represents the capacitance of the vascular bundle and C γ represents the Pith capacitance. Post-processing is conducted on the measured impedance data using four metaheuristic optimization algorithms to extract the parameters of the employed models. The applied algorithms are adopted according to the literature, where they proved to output satisfying results. It is essential to precisely define the factors influencing the optimization algorithm's result to obtain optimal bio-impedance models' parameters. The factors include the objective function, number of search agents, runs and iteration, the upper and lower boundaries, and the vector of optimized variables.
1. The objective function represented in Eq. (6) is the sum of the absolute error between the estimated impedance from the model and the measured impedance of the sample for each frequency point.
where x is the impedance parameters of each model depending on the problem size, Z model (x) is the impedance equation of the models, while Z measured is the measured response of the sample. n is the total number of the measured points. 2. The number of search agents used in the optimization is 60 and runs for 100 independent runs through 1800 iterations for all the tested samples. 3. The search agents search for the best solution in a region defined between a lower (LB) and an upper (UB) boundary defined differently for each model are shown in Table 1.

Experimental results and discussion
In this section, The two proposed stem models are validated by fitting the measured data and using FPA, CS, CSO, and WCA optimization techniques in the models' parameters extraction process. As mentioned in the literature, FPA and CS optimization algorithms are used before in bio-impedance parameter extraction problems and showed good performance, while WCA and CSO are used for the first time for such problems. The four algorithms are compared to select the most fitted algorithm in such a problem by studying the error and convergence curves. Figure 5 shows a flowchart that summarises the employed optimizations; more details about these optimizations can be found in [49][50][51] .    The measured data at a distance = 5 cm and 10 cm for the studied plant samples were fitted on three models (Cole, Double Cole, and Fractional-Order Double shell models) and the two proposed models (Stem and simplified stem models). The extracted models' parameters by WCA are shown in Table 2.
For Marjoram plant samples, the Nyquist plot is plotted for the experimental data and the extracted parameters from the four applied optimization techniques (FPA, CS, CSO, and WCA), as shown in Table 3. The proposed stem model and the simplified model have shown good fitting results. Furthermore, the error between the measured impedance of marjoram by (SP150) and the fitted data is calculated for each model, where the FPA, CS, CSO, and WCA algorithms are applied for each model. WCA algorithm showed the best result compared to FPA, CS, and CSO optimizations in all studied cases. For the Marjoram plant samples at 5 cm distance, the maximum error calculated for Cole-impedance model is around 2%, for Double Cole is 2%, for Double-shell is 0.8%, for the proposed stem model is less than 0.5%, and for the proposed simplified stem model is less than 0.8%. For the case of Marjoram with a distance of 10 cm, the Cole-impedance model has a maximum error greater than 5%, for the Double Cole model is 0.8%, for the Double-shell is greater than 2.5%, while for the proposed model is 0.5%, and for the simplified one is less than 0.4%.
For Salvia plant samples, the Nyquist plot is plotted for the experimental data, and the extracted parameters from the four applied optimization techniques, as shown in Table 4. The proposed stem model and the simplified model have shown good fitting results. Furthermore, the error between the measured impedance of marjoram by (SP150) and the fitted data is calculated for each model, where the four optimization algorithms are applied for each model. WCA algorithm showed the best result compared to FPA, CS, and CSO optimizations in all studied cases. For samples at a 5 cm distance, the maximum error for the Cole model is around 5%, for the Double Cole model is around 1.6%, for the Double-shell model is 1.2%, while for the proposed stem model and the simplified one are less than 0.7%. While for the 10 cm distance, the maximum error for the Cole model is less than 2.5%, for the Double Cole model is equal to 1.3%, for the Double-shell model equals 1.1%, for the proposed model is around 1%, and for the proposed simplified model is around 1.1%.
For Lavandula plant samples, the Nyquist plot is plotted for the experimental data, and the extracted parameters from the four applied optimization techniques as shown in Table 5. The proposed stem model and the simplified model have shown good fitting results. Furthermore, the error between the measured impedance of marjoram by (SP150) and the fitted data is calculated for each model, where the four optimization algorithms are applied for each model. WCA algorithm showed the best result compared to FPA, CS, and CSO optimizations in all studied cases. for samples at 5 cm distance, the maximum error for Cole model is greater than 5.5%, for Double Cole and Double-shell models is 4%, for the proposed stem model is around 1.5%, and for the simplified stem model is around 2%. While for the 10 cm distance, the maximum error for the Cole model is greater than 6%, for the Double Cole model and Double-shell model is 2.5%, for the proposed stem model is around 1%, and for the proposed simplified stem model is around 1.8%.
For more exploration for the performance of the four used optimization algorithms, Convergence curves are investigated at 1800 iteration. Table 6 shows the convergence curves for a sample of Marjoram at 5 cm distance. In all models, WCA optimization converges at 1000 iteration, while CSO optimization shows an inconsistency behaviour. For CS optimization, it converges at 600 iteration for Cole and Double-shell impedance models, and converges at 1500 iteration for the Double Cole and the proposed stem models, while it needs more than 1800 iteration to converge in the proposed simplified stem model. For FPA optimization, Cole impedance model needs 1000 iteration to converge, and around 1700 iteration for Double-shell and the proposed stem model. While Double Cole impedance model converges at 1800 iteration, and the proposed simplified stem model needs more than 1800 iteration to converge. According to this results, WCA optimization outperforms the other three optimizations as it got the lowest error percentage in all cases. FPA and CS optimizations show defects when dealing with a bigger problem size; also, CSO showed an inconsistency behaviour in some cases.
The final outcome is that the proposed stem and the simplified stem models showed a remarkable performance over the commonly used models in plant stem tissues representations. Furthermore, WCA is the recommended technique for bio-impedance problems, especially for the larger problem size.

Conclusion
Two fractional-order bio-impedance models for plant stem characterization were introduced and compared with three known models (Cole, Double Cole, Fractional-Order Double-shell). Their parameters were extracted based on the measured data of three medical plant species using two samples each. The employed optimization algorithms are WCA and CSO optimizations, which are used for the first time in such problem, and compared with two conventional metaheuristic optimization techniques (FPA and CS) used before in similar problems. Error percentage was plotted versus frequency for each model using the four algorithms to test the models' efficiency and the most efficient algorithm for the studied problem. The proposed models showed their significant advantage over the other used models in all electrode positions tested; they give the least fitting error percentage compared with the actual measurements' data. The WCA optimization algorithm demonstrates its accuracy, particularly for a larger problem size where the FPA optimization algorithm shows some defects.