Abstract
Volterra kernel stochastic system identification is a technique that can be used to capture and model nonlinear dynamics in biological systems, including the nonlinear properties of skin during indentation. A high bandwidth and high stroke Lorentz force linear actuator system was developed and used to test the mechanical properties of bulk skin and underlying tissue in vivo using a non-white input force and measuring an output position. These short tests (5 s) were conducted in an indentation configuration normal to the skin surface and in an extension configuration tangent to the skin surface. Volterra kernel solution methods were used including a fast least squares procedure and an orthogonalization solution method. The practical modifications, such as frequency domain filtering, necessary for working with low-pass filtered inputs are also described. A simple linear stochastic system identification technique had a variance accounted for (VAF) of less than 75%. Representations using the first and second Volterra kernels had a much higher VAF (90–97%) as well as a lower Akaike information criteria (AICc) indicating that the Volterra kernel models were more efficient. The experimental second Volterra kernel matches well with results from a dynamic-parameter nonlinearity model with fixed mass as a function of depth as well as stiffness and damping that increase with depth into the skin. A study with 16 subjects showed that the kernel peak values have mean coefficients of variation (CV) that ranged from 3 to 8% and showed that the kernel principal components were correlated with location on the body, subject mass, body mass index (BMI), and gender. These fast and robust methods for Volterra kernel stochastic system identification can be applied to the characterization of biological tissues, diagnosis of skin diseases, and determination of consumer product efficacy.
Similar content being viewed by others
References
Asyali, M. H., and M. Juusola. Use of Meixner functions in estimation of Volterra kernels of nonlinear systems with delay. IEEE Trans. Biomed. Eng. 52(2):229–237, 2005.
Boyer, G., L. Laquièze, A. L. Bot, S. Laquièze, and H. Zahouani. Dynamic indentation on human skin in vivo: aging effects. Skin Res. Technol. 15:55–67, 2009.
Burnham, K. P., and D. R. Anderson. Model Selection and Multi-Model Inference: A Practical Information-Theoretic Approach, 2nd ed. New York: Springer, p. 66, 1998.
Chen, Y., and I. W. Hunter. In vivo characterization of skin using a Wiener nonlinear stochastic system identification method. In: 31st Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 2009, pp. 6010–6013.
Chen, Y., and I. W. Hunter. Stochastic system identification of skin properties: linear and Wiener static nonlinear methods. Ann. Biomed. Eng. 40(10):2277–2291, 2012.
Daly, C. H., and G. F. Odland. Age-related changes in the mechanical properties of human skin. J. Invest. Dermatol. 73:84–87, 1979.
Escoffier, C., J. Rigal, A. Rochefort, R. Vasselet, J. Lévêoque, P. G. Agache. Age-related mechanical properties of human skin: an in vivo study. J. Invest. Dermatol. 93:353–357, 1989.
Flynn, C., A. Taberner, and P. Nielsen. Measurement of the force-displacement response of in vivo human skin under a rich set of deformations. Med. Eng. Phys. 33(5):610–619, 2011.
Friston, K. J., A. Mechelli, R. Turner, and C. J. Price. Nonlinear responses in fMRI: the balloon model, Volterra kernels, and other hemodynamics. NeuroImage. 12:466–477, 2000.
Goussard, Y., W. C. Krenz, L. Stark, and G. Demoment. Practical identification of functional expansions of nonlinear systems submitted to non-Gaussian inputs. Ann. Biomed. Eng. 19:401–427, 1991.
Hayes, W. C., L. M. Keer, G. Herrmann, and L. F. Mockros. A mathematical analysis for indentation tests of articular cartilage. J. Biomech. 5:541–551, 1972.
Hendriks, F. M., D. Brokken, C. W. J. Oomens, D. L. Bader, F. P. T. Baaijens. The relative contributions of different skin layers to the mechanical behavior of human skin in vivo using suction experiments. Med. Eng. Phys. 28:259–266, 2006.
Hendriks, F. M., D. Brokken, J. T. W. M. van Eemeren, C. W. J. Oomens, F. P. Baaijens, and J. B. A. M. Horsten. A numerical-experimental method to characterize the non-linear mechanical behavior of human skin. Skin Res. Technol. 9:274–283, 2003.
Hoffman, A. H., and P. Grigg. Using uniaxial pseudorandom stress stimuli to develop soft tissue constitutive equations. Ann. Biomed. Eng. 30:44–53, 2002.
Hunter, I. W. Measuring properties of an anatomical body. US Patent 7,530,975 B2, 2009.
Hunter, I. W., and Y. Chen. Nonlinear system identification techniques and devices for discovering dynamic and static tissue properties. US Patent Application 20,110,054,354, 2011.
Hunter, I. W., and M. J. Korenberg. The identification of nonlinear biological systems: Wiener and Hammerstein cascade models. Biol. Cybern. 55:135–144, 1986.
Khatyr, F., C. Imberdis, P. Vescovo, D. Varchon, and J. Lagarde. Model of the viscoelastic behavior of skin in vivo and study of anisotropy. Skin Res. Technol. 10:96–103, 2004.
Kibangou, A. Y., and G. Favier. Wiener–Hammerstein systems modeling using diagonal Volterra kernels coefficients. IEEE Signal Process. Lett. 13(6):381–384, 2006.
Korenberg, M. J. Aspects of Time-Varying and Nonlinear Systems Theory with Biological Applications. Montréal, QC: McGill University, 1972.
Korenberg, M. J. Identifying nonlinear difference equation and functional expansion representations: the fast orthogonal algorithm. Ann. Biomed. Eng. 16:123–142, 1988.
Korenberg, M. J. The identification of nonlinear biological systems: Wiener kernel approaches. Ann. Biomed. Eng. 18:629–654, 1990.
Korenberg, M. J., S. B. Bruder, and P. J. McIlroy. Exact orthogonal kernel estimation from finite data records: extending Wiener’s identification of nonlinear systems. Ann. Biomed. Eng. 16:201–214, 1988.
Korenberg, M. J., and I. W. Hunter. The identification of nonlinear biological systems: LNL cascade models. Biol. Cybern. 55:125–134, 1986.
Korenberg, M. J., and I. W. Hunter. The identification of nonlinear biological systems: Volterra kernel approaches. Ann. Biomed. Eng. 24:250–268, 1996.
Korenberg, M. J., and I. W. Hunter. Two methods for identifying Wiener cascades having non-invertible static nonlinearities. Ann. Biomed. Eng. 27:793–804, 1999.
Lee, Y. W., and M. Schetzen. Measurement of Wiener kernels of a non-linear system by cross-correlation. Int. J. Control. 2(3):237–254, 1965.
Ljung, L. System Identification: Theory for the User. Upper Saddle River, NJ: Prentice Hall, 1987.
Marmarelis, P. Z., and K. I. Naka. Nonlinear analysis and synthesis of receptive-field responses in the catfish retina. J. Neurophys. 36:605–648, 1973.
Marmarelis, V. Z. Identification of nonlinear biological systems using Laguerre expansions of kernels. Ann. Biomed. Eng. 21:573–589, 1993.
Marques de Sá, J. P. Applied Statistics: Using SPSS, STATISTICA, and MATLAB. Berlin: Springer, 2003.
Nizet, J. L., C. Piérard-Franchimont, G. E. Piérard. Influence of the body posture and gravitational forces on shear wave propagation in the skin. Dermatology 202:177–180, 2001.
Ottensmeyer, M., and J. Salisbury. In vivo data acquisition instrument for solid organ mechanical property measurement. In: Medical Image Computing and Computer-Assisted Intervention—MICCAI, edited by W.Niessen and M. Viergever, vol. 2208 of Lecture Notes in Computer Science. Berlin: Springer, 2001, pp. 975–982.
Potts, R. O., E. M. Buras, and D. A. Chrisman. Changes with age in the moisture content of human skin. J. Invest. Dermatol. 82:97–100, 1984.
Sanford, E., Y. Chen, I. W. Hunter, G. Hillebrand, and L. Jones. Capturing skin properties from dynamic mechanical analyses. Skin Res. Technol., 2013. doi:10.1111/j.1600-0846.2012.00649.x.
Schetzen, M. Measurement of the kernels of a nonlinear system of finite order. Int. J. Control. 1:251–263, 1965.
Stark, L. W. The pupillary control system: its nonlinear adaptive and stochastic engineering design characteristics. Automatica 5:655–676, 1969.
Tilleman, T. R., M. M. Tilleman, and M. H. Neumann. The elastic properties of cancerous skin: Poisson’s ratio and Young’s modulus. Isr. Med. Assoc. J. 6(12):753–755, 2004.
Timanin, E. M. Interpretation of impedance characteristics of biological soft tissues in the models with a pressure source of vibrations with friction. In: XIII Session of the Russian Acoustical Society, 2003, pp. 581–584.
Tosti, A., G. Compagno, M. L. Fazzini, S. Villardita. A ballistometer for the study of the plasto-elastic properties of skin. J. Invest. Dermatol. 69:315–317, 1977.
Vannah, W. M., and D. S. Childress. Indentor tests and finite element modeling of bulk muscular tissue in vivo. J. Rehabil. Res. Dev. 33(3):239–252, 1996.
Westwick, D. T., and R. E. Kearney. Identification of Nonlinear Physiological Systems. Picscataway, NJ: IEEE Press, 2003.
Wiener, N. Nonlinear Problems in Random Theory. Cambridge, MA: MIT Press, 1958.
Yuan, H., D. T. Westwick, E. P. Ingenito, K. R. Lutchen, and B. Suki. Parametric and nonparametric nonlinear system identification of lung tissue strip mechanics. Ann. Biomed. Eng. 27:548–562, 1999.
Zhang, M., Y. Zheng, and A. F. T. Mak. Estimating the effective Young’s modulus of soft tissues from indentation tests—nonlinear finite element analysis of effects of friction and large deformation. Med. Eng. Phys. 19(6):512–517, 1997.
Zhang, Q., K. R. Lutchen, and B. Suki. A frequency domain approach to nonlinear and structure identification for long memory systems: application to lung mechanics. Ann. Biomed. Eng. 27:1–13, 1999.
Acknowledgments
The authors would like to thank Dr. Lynette Jones, Dr. Cathy Hogan, and Dr. Bryan Ruddy for their advice. This work was supported in part by the National Science Foundation Graduate Research Fellowship.
Author information
Authors and Affiliations
Corresponding author
Additional information
Associate Editor Eiji Tanaka oversaw the review of this article.
Rights and permissions
About this article
Cite this article
Chen, Y., Hunter, I.W. Nonlinear Stochastic System Identification of Skin Using Volterra Kernels. Ann Biomed Eng 41, 847–862 (2013). https://doi.org/10.1007/s10439-012-0726-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10439-012-0726-x