Abstract
We consider direct and inverse problems of three-dimensional quasi-static elastography underlying a cancer diagnosis method. They are based on a model of a tissue exposed to surface compression with deformations obeying linear elasticity laws. The arising three-dimensional displacements of the tissue are described by a boundary value problem for partial differential equations with coefficients determined by a variable Young’s modulus and a constant Poisson ratio. The problem contains a small parameter, so it can be solved using the theory of regular perturbations of partial differential equations. This is the direct problem. The inverse problem is to find the Young modulus distribution from given tissue displacements. A significant increase in Young’s modulus within a certain tissue domain suggests possible malignancy. Under certain assumptions, simple formulas for solving both direct and inverse problems of three-dimensional quasi-static elastography are derived. Three-dimensional inverse test problems are solved numerically with the help of the proposed formulas. The resulting approximate solutions agree fairly well with the exact model solutions. The computations based on the formulas require only several tens of milliseconds on a moderate-performance personal computer for sufficiently fine grids, so the proposed small-parameter approach can be used in real-time cancer diagnosis.
REFERENCES
L. Gao, K. Parker, R. Lerner, et al., “Imaging of the elastic properties of tissue—a review,” Ultrasound Med. Biol. 22, 959–977 (1996).
J. Ophir, S. Alam, B. Garra, et al., “Elastography: Ultrasonic estimation and imaging of the elastic properties of tissues,” Proc. Inst. Mech. Eng. Part H: J. Eng. Med. 213, 203–233 (1999).
J. F. Greenleaf, M. Fatemi, and M. Insana, “Selected methods for imaging elastic properties of biological tissues,” Annu. Rev. Biomed. Eng. 5, 57–78 (2003).
K. J. Parker, L. S. Taylor, S. Gracewski, et al., “A unified view of imaging the elastic properties of tissue,” J. Acoust. Soc. Am. 117, 2705–2712 (2005).
M. Doyley, “Model-based elastography: A survey of approaches to the inverse elasticity problem,” Phys. Med. Biol. R 57, 35–73 (2012).
S. N. Gurbatov, I. Yu. Demin, and N. V. Pronchatov-Rubtsov, Ultrasound Elastography: Analytical Description of Various Modes and Techniques, Physical and Numerical Modeling of Shear Characteristics of Soft Biological Tissues (Nizhegorod. Gos. Univ., Nizhny Novgorod, 2015) [in Russian].
A. A. Oberai, N. H. Gokhale, and G. R. Feijoo, “Solution of inverse problems in elasticity imaging using the adjoint method,” Inverse Probl. 19, 297–313 (2003).
M. Richards, P. Barbone, and A. Oberai, “Quantitative three-dimensional elasticity imaging from quasi-static deformation: A phantom study,” Phys. Med. Biol. 54, 757–779 (2009).
A. S. Leonov, A. N. Sharov, and A. G. Yagola, “A posteriori error estimates for numerical solutions to inverse problems of elastography,” Inverse Probl. Sci. Eng. 25, 114–128 (2017).
A. S. Leonov, A. N. Sharov, and A. G. Yagola, “Solution of the inverse elastography problem for parametric classes of inclusions with a posteriori error estimate,” J. Inverse Ill-Posed Probl. 26, 1–7 (2017).
A. S. Leonov, A. N. Sharov, and A. G. Yagola, “Solution of the three-dimensional inverse elastography problem for parametric classes of inclusions,” Inverse Probl. Sci. Eng. 29 (8), 1055–1069 (2021).
M. Rychagov, W. Khaled, S. Reichling, et al., “Numerical modeling and experimental investigation of biomedical elastographic problem by using plane strain state model,” Fortsch. Akust. 29, 586–589 (2003).
A. S. Leonov, N. N. Nefedov, A. N. Sharov, and A. G. Yagola, “Solution of the two-dimensional inverse problem of quasistatic elastography with the help of the small parameter method,” Comput. Math. Math. Phys. 62 (5), 827–833 (2022).
O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Nauka, Moscow, 1973; Springer-Verlag, New York, 1985).
A. N. Tikhonov, A. B. Vasil’eva, and A. G. Sveshnikov, Differential Equations (Nauka, Moscow, 1980; Springer-Verlag, Berlin, 1985).
V. A. Trenogin, Functional Analysis (Nauka, Moscow, 1980) [in Russian].
A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Nauka, Moscow, 1990; Kluwer Academic, Dordrecht, 1995).
A. S. Leonov, Solution of Ill-Posed Inverse Problems: Theory, Practical Algorithms, and Demonstrations in M-ATLAB (Librokom, Moscow, 2009) [in Russian].
Funding
This work was supported by the Russian Science Foundation, project no. 18-11-00042.
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors declare that they have no conflicts of interest.
Additional information
Translated by I. Ruzanova
Rights and permissions
About this article
Cite this article
Leonov, A.S., Nefedov, N.N., Sharov, A.N. et al. “Fast” Solution of the Three-Dimensional Inverse Problem of Quasi-Static Elastography with the Help of the Small Parameter Method. Comput. Math. and Math. Phys. 63, 425–440 (2023). https://doi.org/10.1134/S0965542523030090
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542523030090