# Non-Invasive and Accurate Blood Glucose Detection Based on an Equivalent Bioimpedance Spectrum
## 7 A Non-Invasive Blood Glucose Detection Model Based on Multiple Regression  
The relationship between the dielectric constant of human tissues and blood glucose concentration was given in (1), and it can be further converted to equivalent biological complex impedance according to (2). Therefore, a regression model can be established to estimate the blood glucose value. However, according to previous studies, the mathematical relationship between blood glucose and complex impedance is not a simple first-order linear relationship. As an example, the complex permittivity of the human cell membrane in the \(\beta\) band follows fourth-order Cole-Cole expansion. The microscopic parameters in fourth-order Cole-Cole expansion and (1) cannot be obtained directly, but the impedance \(Z\), the phase angle \(\varphi\)  
Figure 11: Comprehensive use of the K-nearest neighbor algorithm flow chart. and the resistance \(R\) are positively correlated with the blood glucose concentration \(c_{glu}\). Therefore, these three parameters can replace the microscopic parameters in fourth-order Cole-Cole expansion and (1) in order to effectively estimate the equivalent cell membrane dielectric constant [29; 30], as shown in (5). \[C_{glu}=f(\epsilon^{*})=g(\sum_{i=1}^{4}a_{i}Z^{i}),C_{glu}=f(\epsilon^{*})=g( \sum_{i=1}^{4}b_{i}R^{i}),C_{glu}=f(\epsilon^{*})=g(\sum_{i=1}^{4}c_{i}\varphi^ {i}). \tag{5}\]  
The higher-order multiple regression model is shown in (6). \[y=\beta_{0}+\sum_{i=1}^{4}\beta_{i}X^{i}+\sum_{k=1}^{4}\beta_{k}R^{k}+\sum_{j= 1}^{4}\beta_{j}\varphi^{j}. \tag{6}\]  
Then, the higher-order multiple regression equation can be written as (7) and (8). \[ \begin{split}\bar{y}_{normal}&=-83*Z+0.8*Z^{2}-0.0 4*Z^{3}+18.0*\varphi-25.2*\varphi^{2}-0.07*\varphi^{3}\\ &+162.9*R-2.8*R^{2}+0.02*R^{3},\end{split}  \tag{7}\]  
\[ \begin{split}\bar{y}_{high}&=147.38*Z-1.6*Z^{2}+0.0 8*Z^{3}+184.9*\varphi-404.9*\varphi^{2}+205.1*\varphi^{3}\\ &-3.5*\varphi^{4}+242.8*R+3.9*R^{2}-0.03*R^{3}.\end{split}  \tag{8}\]  
The obtained regression function can be validated through various statistical tests that assess the reliability of the regression model, primarily including goodness-of-fit tests, significance tests, and residual analysis. Among these, the goodness-of-fit test, denoted as \(R^{2}\), reflects the degree of dispersion between the data points of the training sample and the regression function. It is straightforward to calculate and does not require extensive consideration of the model's specific parameters. The formula for this calculation is presented in Formula (9). The range of the goodness-of-fit test \(R^{2}\) is (0, 1), where a larger value indicates a better fit of the regression function. \[R=\sqrt{1-\frac{\sum_{i}^{n}(y_{i}-\bar{y_{i}})^{2}}{\sum_{i}^{n}(y_{i}-\overline {y_{i}})^{2}}}. \tag{9}\]  
The \(F\)-test is a statistical method employed to assess the overall significance of a regression model. It represents the ratio of the mean square regression (MSR) to the mean square error (MSE) of the residuals. This test takes into account the complexity of the model and evaluates the significance of the model as a whole. The specific calculation formula is provided in Formulas (10) and (11). Additionally, a residual analysis is conducted to ascertain whether the residuals follow a normal distribution. \[F=\frac{U/m}{Q/(n-m-1)}\sim F(m,n-m-1), \tag{10}\]  
\[Q=\sum(y_{i}-\widehat{y_{i}})^{2},\quad U=\sum(\widehat{y_{i}}-\overline{y_{i }})^{2}. \tag{11}\]  
The statistical test results of the above higher order multiple regression model are listed in Table 1.  
According to Table 1, the correlation of the standard blood glucose regression model is 0.89, while that of the hyperglycemia regression model is 0.87. Thus, the above regression model is feasible for blood glucose estimation. Furthermore, in the significance test, the \(F\) value of the proposed regression model is greater than that of the lookup table, which proves that the selection of independent variables in the regression model is appropriate.