Supplementary material from "MRI elastography of human intervertebral disc based on an explicit inverse approach"

Intervertebral disc (IVD) degeneration is a significant health issue that can lead to severe complications. Recent research has highlighted the close relationship between disc degeneration and the biomechanical properties of the IVD. This study introduces an innovative approach—magnetic resonance imaging (MRI) elastography of the human IVD—using an explicit inverse solver to identify the non-homogeneous shear modulus map of the IVD. The key advantage of this explicit solver is its streamlined optimization process, focusing only on the shear moduli of the nucleus pulposus (NP), annulus fibrosus (AF), and their interface. This approach reduces the optimization variables, making it more efficient than traditional pixel-based approaches. To validate this method, we conducted a plane strain numerical example, observing a consistent underestimation of the AF/NP shear modulus ratio by a scaling factor of approximately 1.5. Further investigations included comprehensive sensitivity analyses to various noise levels, revealing that the proposed method accurately characterizes shear modulus distribution in the AF and NP regions, with a maximum relative error of the AF/NP shear modulus ratio remaining below 8%. In addition, applying this approach to real human IVDs under in vitro compression or bending, demonstrated its effectiveness, yielding an AF/NP shear modulus ratio within a reasonable range of 6–15. In summary, the proposed method offers a promising direction for MRI elastography of the human IVD.