Prediction of Soft Tissue Viscoelastic Properties by Support Vector Regression

Consider a support vector regression (SVR) model in which the inputs are electrical properties such as the relative permittivity and conductivity. Using these inputs, the model predicts the values of the loss modulus and storage modulus and as a function of frequency.

SVR employs linear functions that are defined in a higher dimensional space for solving the regression problem. In SVR, the estimation is carried out by the minimization of risk using Vapnik’s ^-insensitive loss function (Vapnik et ah, 1996). SVM utilizes a risk function that consists of the empirical error and a regularization term which is obtained from the structural risk minimization principle. Given a set of inputs x„ d_: as the required or the desired value and n as the total number of data patterns, SVMs approximate the function using the following equations (Tay & Cao, 2001):

where q>(x) is the high-dimensional feature space which is nonlinearly mapped from the input space x. The coefficients w and b are estimated by minimizing the regularized risk function given by Equation (2.2).

In the work presented in this section, a radial basis kernel function (Gunn, 1998) is utilized to develop the SVR model.

where p, and p₂ are parameters specified by the user (Kamalanand & Jawahar, 2018).

This presents the results of two different SVR models with a radial basis function kernel of width = 3. In the first case, the SVR model was developed using the values of relative permittivity and conductivity as the inputs and the elastic modulus as the output of the model. Figure 2.8 shows the elastic modulus predicted by the SVR model and the actual values as a function of frequency. In the second case, the SVR model was developed using the frequency-dependent permittivity and conductivity

FIGURE 2.8 Variation of elastic modulus (actual and predicted values using the SVR model) shown as a function of frequency.

as inputs and the viscous modulus as the output of the model. The viscous modulus predicted by the SVR model and the actual values are shown as a function of frequency in Figure 2.9. It is seen that the SVR model is efficient in predicting the values of both elastic and viscous modulus with dielectric properties as the inputs.

In human physiological systems, the soft tissue electrical properties are well correlated with its mechanical properties. In several cases, due to the inaccessibility of tissue for measurement, it is not possible to measure the material properties effectively. The measurement of electrical properties of in-vivo soft tissues is easier compared to the mechanical measurements. Due to the existing correlations between the electrical and mechanical properties of the soft tissues, it is possible to estimate the unknown properties by measurement of few of the mechanical or the electrical properties using computational algorithms.

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