# 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.

# References

Asbach, P., Klatt, D., Hamhaber, U., Braun, J., Somasundaram, R., Hamm, B., & Sack, I. (2008). Assessment of liver viscoelasticity using multifrequency MR elastography. Magnetic Resonance in Medicine, 60(2), 373-379.

Chammas, P., Federspiel, W. J„ & Eisenberg, S. R. (1994). A microcontinuum model of electrokinetic coupling in the extracellular matrix: Perturbation formulation and solution. Journal of Colloid and Interface Science, 168(2), 526-538.

Dasgupta, B. R. & Weitz, D. A. (2005). Microrheology of cross-linked polyacrylamide networks. Physical Review E, 71(2), 021504.

Devi, C. U., Chandran, R. B.. Vasu, R. M.. & Sood. A. K. (2007). Measurement of visco-elastic properties of breast-tissue mimicking materials using diffusing wave spectroscopy. Journal of Biomedical Optics, 12(3), 034035.

Dhar, P. R. & Zu, J. W. (2007). Design of a resonator device for in vivo measurement of regional tissue viscoelasticity. Sensors and Actuators A: Physical, 733(1), 45-54.

Duck, F. A. (2013). Physical properties of tissues: A comprehensive reference hook. Cambridge, MA: Academic Press.

Fatemi, M., Manduca, A., & Greenleaf, J. F. (2003). Imaging elastic properties of biological tissues by low-frequency harmonic vibration. Proceedings of the IEEE, 97(10), 1503-1519.

Foster, K. F. & Schwan, H. P. (1989). Electric properties of tissues and biological materials: A critical review CRC Crit. Annual Review of Biomedical Engineering, 17,25-104. Gabriel, S., Lau, R. W., & Gabriel, C. (1996). The dielectric properties of biological tissues:

III. Parametric models for the dielectric spectrum of tissues. Physics in Medicine and Biology, 4/(11), 2271.

Greenleaf, J. F., Fatemi, M., & Insana, M. (2003). Selected methods for imaging elastic properties of biological tissues. Annual Review of Biomedical Engineering, 5(1), 57-78. Gunn, S. R. (1998). Support vector machines for classification and regression. ISIS Technical Report, 14(1), 5-16.

Howe, R. D., Peine, W. J., Kantarinis, D. A., & Son, J. S. (1995). Remote palpation technology.

IEEE Engineering in Medicine and Biology Magazine, 14(3), 318-323.

Kamalanand, K. & Jawahar, P. (2018). Mathematical modelling of systems and analysis. New Delhi: PHI Learning Pvt. Ltd.

Kamalanand, K., Sridhar, В. T. N.. Rajeshwari, P. M., & Ramakrishnan, S. (2010). Correlation of dielectric permittivity with mechanical properties in soft tissue-mimicking polyacrylamide phantoms. Journal of Mechanics in Medicine and Biology, 10(02), 353-360.

Keshtkar, A., Mesbahi, A., & Mehnati, P. (2008). The effect of bladder volume changes on the measured electrical impedance of the urothelium. International Journal of Biomedical Engineering and Technology, /(3), 287-292.

Klatt, D., Hamhaber, U., Asbach. P., Braun, J., & Sack, I. (2007). Noninvasive assessment of the rheological behavior of human organs using multifrequency MR elastography: A study of brain and liver viscoelasticity. Physics in Medicine and Biology, 52(24), 7281.

Konofagou, E. E„ Ottensmeyer, M., Agabian, S„ Dawson, S. L., & Hynynen, K. (2004). Estimating localized oscillatory tissue motion for assessment of the underlying mechanical modulus. Ultrasonics, 42(1-9), 951-956.

Krishnamurthy. K.. Sridhar, В. T. N.. Rajeshwari, P. M., & Swaminathan, R. (2009). Correlation of electrical impedance with mechanical properties in models of tissue mimicking phantoms. In 13th International Conference on Biomedical Engineering (pp. 1708-1711). Springer, Berlin, Heidelberg.

Kun, S., Ristic, B., Peura, R. A., & Dunn, R. M. (1999). Real-time extraction of tissue impedance model parameters for electrical impedance spectrometer. Medical and Biological Engineering and Computing, 37(4), 428-432.

Lazebnik, M., Converse, M. C., Booske, J. H., & Hagness, S. C. (2006). Ultrawideband temperature-dependent dielectric properties of animal liver tissue in the microwave frequency range. Physics in Medicine and Biology, 5/(7), 1941.

Lazebnik, M., Popovic, D., McCartney, L., Watkins, С. B., Lindstrom, M. J., Harter, J.,... & Temple, W. (2007). A large-scale study of the ultrawideband microwave dielectric properties of normal, benign and malignant breast tissues obtained from cancer surgeries. Physics in Medicine and Biology, 52(20), 6093.

Nash, M. P. & Panfilov, A. V. (2004). Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias. Progress in Biophysics and Molecular Biology, 85(2-3), 501-522.

Novacek, V., Krakovsky, I., Muller, M., & Tonar, Z. (2002). Identification of mechanical parameters of biological tissues. In Proceedings of the Conference Applied Mechanics (pp. 267-272).

O’Rourke, A. P., Lazebnik, M., Bertram, J. M., Converse, M. C., Hagness, S. C., Webster, J. G., & Mahvi, D. M. (2007). Dielectric properties of human normal, malignant and cirrhotic liver tissue: In vivo and ex vivo measurements from 0.5 to 20GHz using a precision open-ended coaxial probe. Physics in Medicine and Biology, 52(15), 4707.

Sarvazyan, A. (1993). Shear acoustic properties of soft biological tissues in medical diagnostics. The Journal of the Acoustical Society of America, 93(4), 2329-2330.

Sierpowska, J., Toyras, J., Hakulinen, M. A., Saarakkala, S., Jurvelin, J. S., & Lappalainen, R. (2003). Electrical and dielectric properties of bovine trabecular bone: Relationships with mechanical properties and mineral density. Physics in Medicine and Biology, 48(6), 775.

Tay, F. E. & Cao, L. (2001). Application of support vector machines in financial time series forecasting. Omega, 29(4), 309-317.

Vapnik, V., Golowich, S. E., & Smola, A. J. (1996). Support vector method for function approximation, regression estimation and signal processing. In Advances in /Veural /nformation Processing Systems (pp. 281-287). doi: 10.5555/2998981.2999021

Whiteley, J. P., Bishop, M. J., & Gavaghan, D. J. (2007). Soft tissue modelling of cardiac fibres for use in coupled mechano-electric simulations. Bulletin of Mathematical Biology, 69(1), 2199-2225.

Woo, S. Y., Gomez, M. A., & Akeson, W. H. (1981). The time and history-dependent viscoelastic properties of the canine medial collateral ligament. Journal of Biomechanical Engineering, /03(4), 293-298.

Youn, J. I., Akkin, T., & Milner, T. E. (2003). Electrokinetic measurement of cartilage using differential phase optical coherence tomography. Physiological Measurement, 25(1), 85.