Linear elasticity

The estimation of elastic properties of tissues can provide medical information, since it is known that tumors may be detected as stiffer regions (e.g. prostate and breast tumors). Ultrasound elastography is an emerging medical imaging modality, that estimates the displacement of tissues under quasi-static loading, or tissues that are crossed by an elastic wave. The estimation of Young's modulus, or the estimation of the shear modulus, is an inverse problem.

Inverse problem for static linear elasticity

The data are the displacement of an elastic medium under a small compression. Only one component of the displacement is observed.

Our contribution: We have proposed an algorithm to estimate the spatial distribution of Young's modulus. It relies on ideas issued from the field of data assimilation and optimal control. The cost function is minimized using a Gauss-Newton algorithm, using direct and adjoint differentiation.

Results with synthetic data:


Left: true inclusion. Right: reconstructed Young's modulus (2% noise added)

Result of the inversion with experimental data (coll. R. Souchon, Inserm U556):

Elastographic data are obtained using a modified endorectal probe. This allows to estimate radial displacements of a hollow cylindrical agar phantom. The phantom contains 6 inclusions of widths ranging from 0.55 to 2.6mm (phantom not displayed on the picture). The inversion procedure allows to localize 4 out of 6 inclusions, and provides a quantitative estimation of the relative stiffness of the inclusions.


Left: experimental setup; center: strain elastogram (in %); right: reconstructed Young's modulus (relative units)

More details:

Relative Young's modulus identification using elastography

Detection of small inclusions using elastography

Inverse problem for the wave equation

The scalar wave equation describes the propagation of a shear wave in an incompressible medium.

Our contribution: We proposed a variational data assimilation method to reconstruct the velocity of the wave. This algorithm proved to be relatively robust to noise.

Results for synthetic data of a plane wave hitting two stiffer inclusions.


From left to right: true velocity; reconstructed velocity with 5% added noise; 10% added noise; 20% added noise.

More details

Variational data assimilation to estimate the velocity in the wave equation

Linear elasticity

Inverse problem for static linear elasticity

Results with synthetic data:

Left: true inclusion. Right: reconstructed Young's modulus (2% noise added)

Result of the inversion with experimental data (coll. R. Souchon, Inserm U556):

Left: experimental setup; center: strain elastogram (in %); right: reconstructed Young's modulus (relative units)

Inverse problem for the wave equation

Results for synthetic data of a plane wave hitting two stiffer inclusions.

From left to right: true velocity; reconstructed velocity with 5% added noise; 10% added noise; 20% added noise.

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