2010 Volume 31 Issue 5 Pages 347-359
We previously reported several useful shear modulus reconstruction methods obtained from motion or static equations and reference shear moduli that yield an absolute reconstruction of shear modulus, density or inertia using the finite difference method (FDM) or finite element method (FEM) with strain tensor measurement (i.e., methods referred to as Methods A to C and F). Methods A to C are particularly useful for three-dimensional (3D) shear modulus reconstruction in an incompressible case or in the case of an inhomogeneous Poisson’s ratio by using the mean normal stress as an unknown, whereas Method F using a typical Poisson’s ratio is particularly useful when the reconstruction dimensionality is low or when the Poisson’s ratio is homogeneous. For instance, when using FDM, although Method B permits the absolute reconstruction of a mean normal stress by using a reference mean normal stress, Methods A and C yield biased mean normal stress reconstructions using no reference value and a quasi-reference value (e.g., zero), respectively. In this report, we describe our newly developed Methods D and E obtained from Methods A to C and F, which yield a unique relative shear modulus reconstruction with no geometrical artifact. In Method D, no reference shear modulus is used with Methods A to C and F, whereas in Method E, an arbitrary finite value (e.g., unity) is used with Methods A to C and F as a quasi-reference shear modulus at one or more arbitrary points (i.e., a quasi-reference point) or in an arbitrary region (i.e., a quasi-reference region) in the region of interest. That is, both Method D and Method E permit the visualization of the shear modulus distribution without using an absolute reference value (e.g., no reference material or typical value), which is particularly useful when dealing with deeply located tissues such as the liver and heart tissues. Methods D and E use iterative methods such as the conjugate gradient method to solve algebraic equations. As confirmed from simulated phantom deformation data, each version of Method E results in a higher convergence speed and reconstruction accuracy than the corresponding Method D. Although several versions of Method D using FEM are practically useful, Method E obtainable should be used for such relative reconstructions rather than the corresponding version of Method D, in combination with Methods A to C and F.
