Reconstruction of nonuniform residual stress for soft hyperelastic tissue via inverse spectral techniques

Residual stress is a stress that can exist in a body in the absence of externally applied loads. It is a challenge to model and predict it in soft tissues both theoretically and experimentally. Mathematical inverse spectral techniques are developed in this article to reconstruct the residual stress in the arterial wall. The techniques are theoretically based on a novel use of the intravascular ultrasound technology (IVUS) by obtaining several natural frequencies of the vessel wall material. As the IVUS is interrogating inside the artery, it produces small amplitude, high frequency time harmonic vibration superimposed on the quasi-static deformation of the pre-stressed artery resulting from pulsatile blood flow. Accordingly, two categories of boundary value problems are formulated to form Sturm–Liouville problems (SLP) with the natural eigenfrequencies from IVUS implementation as eigenvalues of the SLP. Via an optimization approach instead of the traditional equation-solving method, the algorithms for an inverse spectral technique are established to recover the residual stress as functions. Robustness of the algorithms is increased by overestimation of the problem. Comprehensive tests are performed to assess the accuracy of the solution. Numerical examples are displayed to illustrate the application of the technique.