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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 underin vitrocompression 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. 

Based on the Hirota bilinear form of the (2+1)-dimensional Ito equation, one class of lump solutions and two classes of interaction solutions between lumps and line solitons are generated through analysis and symbolic computations with Maple. Analyticity is naturally guaranteed for the presented lump and interaction solutions, and the interaction solutions reduce to lumps (or line solitons) while the hyperbolic-cosine (or the quadratic function) disappears. Three-dimensional plots and contour plots are made for two specific examples of the resulting interaction solutions. 

Based on the Hirota bilinear form of the (2 + 1)-dimensional Ito equation, one class of lump solutions and two classes of interaction solutions between lumps and line solitons are generated through analysis and symbolic computations with Maple. Analyticity is naturally guaranteed for the presented lump and interaction solutions, and the interaction solutions reduce to lumps (or line solitons) while the hyperboliccosine (or the quadratic function) disappears. Three-dimensional plots and contour plots are made for two specific examples of the resulting interaction solutions. 

A series of Ag( i ) and Cu( i ) complexes [Ag 3 (L 1 ) 2 ][PF 6 ] 3 ( 8 ), [Ag 3 (L 2 ) 2 ][PF 6 ] 3 ( 9 ), [Cu(L 1 )][PF 6 ] ( 10 ) and [Cu(L 2 )][PF 6 ] ( 11 ) have been synthesized by reactions of the tridentate amine-bis(N-heterocyclic carbene) ligand precursors [H 2 L 1 ][PF 6 ] 2 ( 6 ) and [H 2 L 2 ][PF 6 ] 2 ( 7 ) with Ag 2 O and Cu 2 O, respectively. Complexes 10 and 11 can also be obtained by transmetalation of 8 and 9 , respectively, with 3.0 equiv. of CuCl. A heterometallic Cu/Ag–NHC complex [Cu 2 Ag(L 1 ) 2 (CH 3 CN) 2 ][PF 6 ] 3 ( 12 ) is formed by the reaction of 8 with 2.0 equiv. of CuCl. All complexes have been characterized by NMR, electrospray ionization mass spectrometry (ESI-MS), and single-crystal X-ray diffraction studies. The luminescence properties of 10–12 in solution and the solid state have been studied. At room temperature, 10–12 exhibit evident luminescence in solution and the solid state. The emission wavelengths are found to be identical at 483 nm in CH 3 CN, but they are 484, 480 and 592 nm in the solid state for 10–12 , respectively. These results suggest that 12 dissociates into two molecules of 10 and Ag( i ) ions in solution. Complex 12 is the first luminescent heterometallic Cu/Ag–NHC complex.