Chemicals and container

Glucose used in the assays was purchased from Merck/Germany. Mica glass (100 mL total volume) was used for all assays. Each experiments were carried out with 50 mL working volume and they were performed as triplicate to increase the repeatability of the results.

Calculation of the scattering parameters

When characterizing transmission lines, the scattering parameters of the transmission line are measured. The scattering parameter quantizes the parts of an incoming electromagnetic wave that are transmitted and reflected by a network. Scattering parameters (S) are a useful technique which shows the circuit in a matrix without knowing which circuit elements a wave circuit consists of, without needing to know its property or internal property. S₁₁ (reflection coefficient) defines the proportion of the amplitude of the reflected signal to the amplitude of the transmitted signal. Hence, in the present study, S₁₁ was calculated using the following Equation 1 and 2 [Reference Pozar24]. Eventually, measured magnitude values (y) were transformed to decibel values (dB) by Equation 3.

(1)\begin{equation}\left[ {\begin{array}{*{20}{c}} {\mathop V\nolimits_1^ - } \\ {\mathop V\nolimits_2^ - } \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\mathop S\nolimits_{11} }&{\mathop S\nolimits_{12} } \\ {\mathop S\nolimits_{21} }&{\mathop S\nolimits_{22} } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\mathop V\nolimits_1^ + } \\ {\mathop V\nolimits_2^ + } \end{array}} \right]\end{equation}

(2)\begin{equation}\left. \begin{gathered} \mathop S\nolimits_{11} = \frac{{\mathop V\nolimits_1^ - }}{{\mathop V\nolimits_1^ + }}\left| {\mathop V\nolimits_2^ + = 0,} \right.\mathop S\nolimits_{12} = \frac{{\mathop V\nolimits_1^ - }}{{\mathop V\nolimits_2^ + }}\left| {\mathop V\nolimits_1^ + = 0} \right. \hfill \\ \mathop S\nolimits_{21} = \frac{{\mathop V\nolimits_2^ - }}{{\mathop V\nolimits_1^ + }}\left| {\mathop V\nolimits_2^ + = 0,} \right.\mathop S\nolimits_{22} = \frac{{\mathop V\nolimits_2^ - }}{{\mathop V\nolimits_2^ + }}\left| {\mathop V\nolimits_1^ + = 0} \right. \hfill \\ \end{gathered} \right\}\end{equation}

(3)\begin{equation}y(dB) = 20 \cdot \log _{10}^{(y)}\end{equation}

Sample preparation and vector network analyzer

In the first part of the experiment, stock glucose solution (200 g/L) was prepared in a 50 mL sterile falcon tube. This solution was transferred to a mica glass container after measuring the dB value of the empty mica glass as a blank sample. After that step, glucose solution was serially diluted with distilled water up to 1 g/L glucose. The dilution step size was set to 1 g/L for each dilution. In the last dilution of this part, only distilled water (50 mL) was measured, making a total of 200 measurements.

In the second step, 20 000 mg/L (20 g/L) stock glucose solution was prepared in a 50 mL sterile falcon tube. This solution was transferred to a mica glass container after measuring the dB value of empty mica glass as a blank sample. The glucose solution was serially diluted with distilled water up to 15 000 mg/L (15 g/L) glucose. The dilution step size was set to 25 mg/L for each dilution. Distilled water (50 mL) was also measured, making a total of 200 measurements. To ensure reproducibility, during the experiments, the container was not moved, and all dilutions were performed with using micropipettes.

VNA (Rohde Schwarz/ZNB40-B22/Germany) equipped with WR-28 adapter and coaxial cable (miniband K-10) was used for experiments. S ₁₁ parameter was monitored between 20 and 40 GHz. All equipment and adapter were calibrated for 50 Ω impedance. All assays were performed at 25^°C and glucose solutions were measured five times using the VNA. Afterwards, average spectra were analyzed to determine the differentiation of glucose concentrations. The actual experimental setup was shown in Fig. 1.

Figure 1. (a) Experimental setup for VNA-based microwave measurement of increased glucose concentrations between 20 and 40 GHz (b) Schematic representation of system and experimental approaches (T: 25°C, number of point: 2000, sweep: 10).

Estimation of glucose level with machine learning methods

Regression is a technique that is used in data science and statistics to model the relationship between a dependent variable and one or more independent variables. The main purpose of this analysis is to predict how the dependent variable changes depending on the independent variables and to understand this change. In the present study, the estimation of glucose levels is performed with three different regression methods, namely DT, RF, and SVR. The employed methods are explained in the following subsections, respectively.

Decision tree

The Decision tree (DT) is a supervised learning algorithm with a tree-like structure. It consists of root node, decision nodes (branches), and leaf node units. The root node is the initial node of the tree where the data set is split according to specific criteria, the split continues with the decision nodes and ends at the leaf nodes. In DT regression, the value at the terminal nodes expresses the average response of the observations that fall within the specific region during training. Consequently, when a new, unseen observation is encountered, the model predicts the response by using the average value of the corresponding region [Reference Czajkowski and Kretowski25]. As can be seen in Fig. 2, the DT makes decisions by dividing nodes into sub-nodes. This subdivision continues throughout the training process until only homogeneous nodes remain [Reference Sneha and Gangil26]. The prediction of the ${i^{th}}$ sample $\hat y{}_i$ in the test data set is made by averaging the samples in the leaf node.

Figure 2. Structure of DT.

Random forest

Random forest (RF) algorithm is an ensemble learning method that constructs a multitude of decision trees for classification or regression tasks. In this method, the bagging technique is employed to construct multiple decision trees that operate in parallel and independently. The bagging technique involves bootstrapping and aggregation. Bootstrapping is the process of creating various subsets of the training data with replacement. DT regression models are created with these data subsets. In the aggregation process, the result of each DT regression model is found and averaged to determine the final estimation. In the RF method with M decision trees, the prediction result is calculated as given in Equation 4.

(4)\begin{equation}\overset{\text{$\smash{\displaystyle\frown}$}}{{f} _{rf}^N} = \frac{1}{N}\mathop \sum \limits_{n = 1}^N T(x)\end{equation}

The structure of the RF method is shown in Fig. 3.

Figure 3. Visual illustration of RF.

Support vector regression

Support vector machines (SVM) is a learning method developed by Vapnik and used for classification and regression problems [Reference Sain and Vapnik27, Reference Smola and Schölkopf28].

Let $\left\{ {\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right), \ldots ,\left( {{x_n},{y_n}} \right)} \right\} \subset \chi \times \mathbf{R}$ be the training data, where $\chi $ denotes the space of input patterns. The goal is to obtain a function $f(x)$ that has the maximum deviation $\varepsilon $ from the targets ${y_i}$ obtained for all training data and is also as smooth as possible. In other words, errors smaller than $\varepsilon $ are ignored, while a deviation larger than $\varepsilon $ is not accepted.

The general mathematical form of the support vector regression (SVR) defined as in Equation 5.

(5)\begin{equation}f\left( x \right) = w \cdot \varphi \left( x \right) + b\end{equation}

where $\varphi \left( x \right)$ denotes the kernel function, $w$ corresponds to the weights, b is the bias. In the case of Equation 5, flatness means looking for a small $w$. One way to achieve this is to minimize the norm, ${\left\| w \right\|^2} = \left\langle {w,w} \right\rangle $. The present problem can therefore be written as a convex optimization problem as in Equation 6.

(6)\begin{equation}\begin{gathered} minimize\,\,\frac{1}{2}{\left\| w \right\|^2} \hfill \\ subject\,to: \hfill \\ \,\,\,\,\,\,{y_i} - \langle w,\varphi \left( {{x_i}} \right)\rangle - b \leq \varepsilon \hfill \\ \,\,\,\,\,\,\langle w,\varphi \left( {{x_i}} \right)\rangle + b - {y_i} \leq \varepsilon \hfill \\ \end{gathered} \end{equation}

The assumption in Equation 6 implies that such a function $f$ exists that approximates all pairs ( ${x_i},{y_i}$) with $\varepsilon $ precision. This assumption also says that the convex optimization problem is feasible. But in some cases we may want to allow some errors. For this, one can add slack variables ${\xi _i},\xi _i^*$ to deal with the infeasible constraints of the optimization problem. This leads to the equation given by [Reference Sain and Vapnik27].

(7)\begin{equation}\begin{gathered} minimize\,\,\frac{1}{2}{\left\| w \right\|^2} + C\sum\limits_{i = 1}^n {\left( {{\xi _i} + \xi _i^*} \right)} \hfill \\ subject\,to: \hfill \\ \,\,\,\,\,\left[ {w \cdot \varphi \left( {{x_i}} \right) + b} \right] - {y_i} \leq \varepsilon + \,\xi _i^* \hfill \\ \,\,\,\,\,{y_i} - \left[ {w \cdot \varphi \left( {{x_i}} \right) + b} \right] \leq \varepsilon + \,{\xi _i} \hfill \\ \,\,\,\,\,{\xi _i},\xi _i^* \geq 0 \hfill \\ \end{gathered} \end{equation}

Different kernel functions are proposed to improve the performance of SVR. Commonly used kernel functions include the linear, polynomial, radial basis function (RBF), and sigmoid kernels.

A graphical illustration of the SVR conversion process is presented in Fig. 4. The samples in the original input space (low dimensional) are mapped to a feature space (high dimensional) through the non-linear mapping function $\varphi \left( x \right)$. Data points located on or outside the $\varepsilon $−tube of the decision function are defined as support vectors and are marked with red stars. The right side of Fig. 4 shows the $\varepsilon $−insensitive loss function, where ${\xi _i}$ and $\xi _i^*$ are slack variables are employed to handle the permissible positive or negative errors.

Figure 4. Illustration of the mapping input space × into high-dimensional feature space and the soft margin loss setting for a SVR.

Performance metrics

In artificial intelligence problems, performance metrics are used to measure the success of the models and compare them with each other. To calculate the performance of the machine learning method, the data set is divided into two parts: training and test sets. While the machine learning model is created with the training data, the performance of this model is calculated using the samples from the test data. Different performance measures have been developed in the literature based on predictive modeling techniques. Commonly utilized performance metrics for regression problems are MAE, MSE, RMSE, and R ².

MAE is determined by taking the average of the absolute differences between the observed values and the predicted values. MAE is defined as

(8)\begin{equation}MAE = \frac{{\sum\limits_{i = 1}^N {\left| {{y_i} - \hat y{}_i} \right|} }}{N}\end{equation}

where ${y_i}$ corresponds to the ${i^{th}}$ observed value, $\hat y{}_i$ is the ${i^{th}}$ predicted value, and N equals the number of samples in the data set.

MSE is calculated as the mean square of the difference between the observed value and the predicted value. MSE emphasizes larger errors due to the presence of the squared term in the equation. MSE is calculated as

(9)\begin{equation}MSE = \frac{{\sum\limits_{i = 1}^N {{{\left( {{y_i} - \hat y{}_i} \right)}^2}} }}{N}\end{equation}

RMSE is simply equal to the square root of MSE. RMSE is computed by taking the square root of average squared differences between the predicted and actual values. Because RMSE calculates the square root of the average squared errors, it brings the error metric back to the original scale of the target variable. RMSE is computed as

(10)\begin{equation}RMSE = \sqrt {\frac{{\sum\limits_{i = 1}^N {{{\left( {{y_i} - \hat y{}_i} \right)}^2}} }}{N}} \end{equation}

R ², also known as the coefficient of determination, indicates the proportion of the variance in the target variable explained by the predictions of the regression model [Reference Isaac and Saha29]. R ² is defined as

(11)\begin{equation}{R^2} = 1 - \frac{{\sum\limits_{i = 1}^N {{{\left( {{y_i} - \hat y{}_i} \right)}^2}} }}{{\sum\limits_{i = 1}^N {{{\left( {{y_i} - \overline y } \right)}^2}} }}\end{equation}

where $\hat y{}_i$ equals ${i^{th}}$ observed value, $\hat y{}_i$ is the ${i^{th}}$ predicted value, and $\overline y $ corresponds to the mean of observed values. R ² is also named as coefficient of determination, it can take values between 0 and 1. As the R ² score approaches 1, it indicates that the model fits better.

K-fold cross-validation

To evaluate the performance of the models, the data set is divided into training and test sets. The machine learning model is created with the training set, and the performance of the model is evaluated on the test set. When the data set is limited, separating the data set into training and test sets is insufficient to evaluate the performance of the model. In this case, a different approach called k-fold cross-validation is often used. In k-fold cross-validation, the data set is randomly divided into k equal parts. One part is reserved for testing, while remaining k − 1 parts are used for training. This process is iterated for k times, and the results of these iterations are then averaged. In this way, all samples in the data set are used for both training and testing of the model. A sample illustration of k-fold cross-validation for a k value of 5 is shown in Fig. 5.

Figure 5. K-fold cross-validation for k = 5.

Experimental results

In this study, a new machine learning model-based system was established to measure glucose concentration in distilled water. Initially, measurements were taken from the VNA via the WR-28 adapter and glucose level estimation was performed with three different machine learning methods: DT, RF, and SVR. In addition, the performances of these methods were benchmarked. All of these methods take method-specific hyperparameters, and these parameters need to be optimized according to the data set to be applied. In order to obtain the best performance for each compared method, a grid search method was applied to the method-specific hyperparameters. In the grid search technique, the performance of the method was measured separately using each pair of hyperparameters and the hyperparameters that give the best score in the calculated performance metrics determine the final performance of the method. The search space of the hyperparameters for the compared methods are given in Table 1. In addition, k value was chosen as 10 in the k-fold cross validation technique and it is repeated 10 times with random shuffling in each iteration to increase reliability of the results.

Table 1. Hyperparameter search space of the methods

Analysis of the stem on gram scale

The first data set is prepared with the increment of glucose in gram scale starting from 1 to 200. In this data set, the results in the R ² performance metric of the three compared methods are approximately close to each other and take values around 0.99 (Table 2). This high value of R ² (0.99) indicates that the predictions of the model are very close to the measurements taken. However, an in-depth analysis of the benchmarked methods on a different performance metric would be more accurate for evaluating the success of the compared methods and determining the best method. For this reason, a detailed performance analysis was performed using RMSE, another regression performance metric frequently used in the literature. Figure 6 visualizes the RMSE scores for the performance of the models in the grid search range for the benchmarked methods.

Figure 6. RMSE score of the benchmarked methods on gram scale analysis (a) DT, (b) RF, (c) SVR-linear, and (d) SVR-RBF.

Table 2. The best scores obtained in benchmarked methods in gram scale

The RMSE scores of the DT method over the specified range are shown in Fig. 6(a). Changing the minimum_sample_split hyperparameter slightly changes the RMSE scores, provided that the maximum depth hyperparameter remains the same. However, RMSE scores increase substantially when the maximum depth hyperparameter falls below 4. Figure 6(b) presents the RMSE scores of the RF method. In the RF method, the RMSE results are very high for values of the number_of_tree hyperparameter up to 4. For values of the number_of_tree hyperparameters greater than 4, the RMSE results are satisfactory and vary within a limited range. As can be seen from Fig. 6(b), the effect of the maximum_depth hyperparameter is quite limited compared to the number_of_trees. The performance of the SVR with linear kernel (SVR-Linear) is shown in Fig. 6(c). In SVR-linear, the variation of RMSE scores in the grid search space is rather restricted. The difference between the highest (4.3036 mg/dL) and lowest RMSE scores (2.7976 mg/dL) is 1.5060. Figure 6(d) visualizes the RMSE scores obtained with the RBF kernel of the SVR (SVR-RBF) method. In SVM-RBF, the best scores are obtained in the range 2⁻¹⁰ − 2⁻⁸ for $\gamma $ and 2³ − 2¹³ for Cost (C). Outside of this range, the performance gradually decreases. The best RMSE scores achieved in each benchmarked method are given in Table 2. Based on the results in Table 2, the best RMSE scores in all performance metrics were obtained in the RF method with hyperparameters number_of_trees = 16, maximum_depth = 16.

Analysis of the stem on milligram scale

In the first part of the study, glucose concentrations between 0 and 200 g/L with 1 g/L intervals were measured via VNA in order to apply machine learning methods. According to the experiments, high R ² and low RMSE performance results were obtained in the benchmarked methods. Thus, during the second step of the experiments, sensitivity of the current measurement method was tested for glucose concentrations between 15 000 and 20 000 mg/L with 25 mg/L intervals. Hence, it was aimed to evaluate the efficiency of the system for highly sensitive analyzes such as blood glucose levels. As previously mentioned for the gram-precision prepared data set, the R ² performance scores of the benchmarking methods in the milligram-precision prepared data set are very close to each other and range between 0.9812 and 0.9923 (Table 3). Therefore, the RMSE performance scores of the benchmarked methods were analyzed in detail. Figure 7 shows the results of DT, RF, SVR-linear, and SVR-RBF methods on the RMSE metric.

Figure 7. RMSE score of the benchmarked methods on milligram-scale analysis (a) DT, (b) RF, (c) SVR-linear, and (d) SVR-RBF.

Table 3. Lowest performances obtained in benchmarked methods in milligram-scale

The performance of the DT method is shown in Fig. 7(a). For the same minimum_sample_split hyperparameter, the effect of changing the maximum_depth hyperparameter on the RMSE scores of the method is quite limited. For the maximum_depth hyperparameter in the range 4 to 25 for all values of the minimum_sample_split, the performance is almost unchanged. At maximum_ depth hyperparameter values lower than 4, performance decreases gradually. The performance of the RF method is presented in Fig. 7(b). The performance of the RF method is almost the same in the range of number_of_trees 2⁵ − 2¹⁰ and maximum_depth 4 − 32 (obtains the best performance), but outside this range the RMSE values gradually increase and therefore the performance decreases.

The variation of the RMSE values in the best performance range is between 1.1119 mg/dL and 1.2449 mg/dL and the variation in this range is approximately 12%. The observation of such a variation (12%) over a certain range in the same method shows the importance of adjusting the method-specific hyperparameters. The RMSE scores of the SVR-linear method in the specified range are shown in Fig. 7(c). As in the DT method Fig. 7(a), the variation of the C hyperparameter in SVR-linear was rather limited when the $\varepsilon $ hyperparameter remained constant. When the hyperparameter of $\varepsilon $ is greater than 2⁻³, the performance of SVR-linear drops significantly. Figure 7(d) shows the RMSE scores of the SVR-RBF method. Similar to the result of the SVM linear method, the effect of the hyperparameter C is fairly low. The performance of the method decreases significantly and gradually as the $\gamma $ value increases from 2⁻⁸. Table 3 shows the lowest scores obtained by the compared methods on all performance metrics in the milligram-scale data set and the hyperparameters at which these scores were obtained. As with the gram-scale data set, the lowest scores were obtained in the RF method with the hyperparameters number of trees = 2⁸ and maximum depth = 30 in the mg-scale data set.

To estimate glucose concentration non-invasively, we employed three machine learning algorithms: DT, RF, an SVR. These models were chosen due to their proven effectiveness in nonlinear regression tasks and ability to handle multivariate data [Reference Smola and Schölkopf28, Reference Breiman30, Reference Quinlan31]. The input features for the models are derived from the measured S-parameters obtained via a VNA, which capture the electromagnetic response of the sensing system influenced by glucose concentration. DT models learn a hierarchy of rules based on these features, allowing for interpretable decision-making but may suffer from overfitting in complex datasets [Reference Breiman30]. RF, as an ensemble of DT, mitigates overfitting by averaging predictions from multiple trees trained on random subsets of the data, increasing robustness and generalization [Reference Quinlan31]. SVR constructs a regression function in a high-dimensional feature space using kernel functions, making it particularly suitable for capturing the nonlinear relationship between VNA-derived features and glucose levels [Reference Smola and Schölkopf28]. These models were trained and evaluated using cross-validation to ensure reliable performance estimation and avoid bias. Their predictions were compared to determine the most accurate and robust approach for non-invasive glucose estimation.

Comparison of the system

Details of the studies in the literature in which glucose concentration was estimated by non-invasive measurements are given in Table 4. As can be seen from Table 4, the performance of the proposed system is significantly superior when compared to some of the previous reports about glucose estimation studies in the literature. Among the previous works, the highest R ² scores were obtained as 0.96, 0.95, and 0.99 from [Reference Naresh, Nagaraju, Kollem, Kumar and Peddakrishna23, Reference Anupongongarch, Kaewgun, O’reilly and Khaomek32, Reference Larin, Eledrisi, Motamedi and Esenaliev33], respectively, which are quite similar to those obtained from the current study. This R ² scores observed from methods [Reference Naresh, Nagaraju, Kollem, Kumar and Peddakrishna23] and the proposed system indicates that 99% of the variance in the dependent variable is explained by the independent variables. Thus, it is more appropriate to perform the comparison on the RMSE metric, which is frequently used in comparing regression methods. On the other hand, the lowest RMSE scores among the values given in the literature were calculated as 5.61 mg/dL, 3.02 mg/dL, and 5.53 mg/dL in [Reference Naresh, Nagaraju, Kollem, Kumar and Peddakrishna23, Reference Xiao, Yu, Li, Song and Kikkawa34, Reference Jain, Maddila and Joshi35] (2019), respectively. Among the studies in Table 4, the lowest RMSE score was calculated as 3.02 mg/dL which was found as only 1.11 mg/dL in the current study. When the proposed system was compared with the closest studies, it was observed that the RMSE scores were 63.18%, 79.93%, and 80.22% lower.

Table 4. Performance comparison of the current work with existing studies

The results obtained from current study can be comparable with other re- ports in the literature. For example, Naresh et al. [Reference Naresh, Nagaraju, Kollem, Kumar and Peddakrishna23] proposed a dual wavelength optical system using 950 nm and 940 nm to detect the glucose non- invasively with 575 samples (460 of them were allocated for training and 115 for testing). R ², MSE, MAE, and RMSE values were observed as 0.99, 9.16 (mg/dL)², 2.49 mg/dL, and 3.02 mg/dL, respectively. Furthermore, Jain et al. [Reference Jain, Maddila and Joshi35] tested a non-invasive glucose detection system using near infrared spectroscopy (NIR) absorbance and reflectance spectroscopy technique. As a result of the study with 25 subjects, R ², MAE, and RMSE values were found as 0.908, 3.87 mg/dL and 5.61 mg/dL, respectively. In an another report, photoacoustic measurements were taken on glucose solutions ranged from 0 to 500 mg/dL. Calibration of photoacoustic measurements from solutions by applying Gaussian kernel-based regression resulted in RMSE, mean absolute relative difference (MARD), and mean absolute difference (MAD) of 7.64 mg/dL, 2.07% and 5.23 mg/dL, respectively [Reference Pai, De and Banerjee36]. On the other hand, in the current study, R², MSE, MAE, and RMSE values were obtained as 0.9932, 1.28 (mg/dL)², 0.87 mg/dL, and 1.11 mg/dL, respectively. In comparison with the existing studies in the literature, the presented studies resulted in significant improvement (63%) in terms of RMSE performance metrics. This enhancement can be related to several factors. First, hyperparameter tuning of the benchmarked models are made to obtain he highest scores in each method. Second, in the current study, 400 samples in total were used for machine learning. Usage of such a high number of samples can lead to improved accuracy and generalization of the measurements, better representation of the data distribution, reduced bias. By this way, it was possible to obtain more reliable metrics. Third, serial dilution of glucose during experiments may have provided a stable environment which support the repeatability.