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Elastography is developed as a quantitative approach to imaging linear elastic properties of tissues to detect suspicious tumors. In this paper a nonlinear elastography method is introduced for reconstruction of complex breast tissue properties. The elastic parameters are estimated by optimally minimizing the difference between the computed forces and experimental measures. A nonlinear adjoint method is derived to calculate the gradient of the objective function, which significantly enhances the numerical efficiency and stability. Simulations are conducted on a three-dimensional heterogeneous breast phantom extracting from real imaging including fatty tissue, glandular tissue, and tumors. An exponential-form of nonlinear material model is applied. The effect of noise is taken into account. Results demonstrate that the proposed nonlinear method opens the door toward nonlinear elastography and provides guidelines for future development and clinical application in breast cancer study.

Breast cancer is one of the major threats to public health all over the world. Currently, X-ray mammography is the primary method for early detection and characterization of breast tumors [

To solve these problems associated with mammography, a number of technologies have been explored. Detection and characterization of breast tumors can be enhanced by recognizing the difference of elastic modulus (stiffness) among normal soft tissues and malignant and benign tumors. Elastic properties of breast tissues may become an indicator of histological diagnosis [

The present work aims at developing a nonlinear elastography model for breast tissues. We introduce a nonlinear adjoint gradient method that significantly improves the numerical efficiency and enhances the stability of elastography reconstructions. In Section

We use continuum description for the breast tissues. Let

Experimental measurements for elastography include displacement and force. We consider that the biological object

For elastography problem, the obtained

Efficient and robust optimization-based elastography schemes request user-supplied gradient

As shown in Figure

Overall flowchart for nonlinear reconstruction of material parameters

This section presents phantom simulations to identify the nonlinear elastic properties of the fatty, glandular, and cancerous tissues in a breast. First of all, a 3D breast FEM phantom attracting from the real data is introduced. Fung’s model [

To perform numerical simulations, a 3D numerical heterogeneous breast phantom extracting from real CT images, containing fatty tissue, glandular tissue, and a tumor is established (Figure

A 3D heterogeneous breast phantom extracted from real CT images, containing fatty tissue, glandular tissue, and a tumor.

The phantom is discretized with standard 3D tetrahedral elements, consisting of 7303 elements and 1583 nodes.

It is well known that the mechanical behavior of biological soft tissue is nonlinear. Hyperelastic models have commonly been used to represent the stress-strain relation of biological soft tissue [

axial stress

Equation (

Nonlinear stress-strain curves for six breast tissues (fatty tissue, glandular tissue, lobular carcinoma, fibroadenoma, infiltrating ductal carcinoma, and ductal carcinoma in situ). (Figure redrawn based on [

In this study, exponential hyperelastic model is used and material parameters are regressed from Wellman [

After the breast phantom and material model are established, a forward problem is solved in which material parameters and external loading are given and deformation is solved. Displacements

Titled compression is given by two paddles. The one close to tumor gives displacement loading on breast phantom and another is fixed during loading.

Four types of compression loading by paddles. Blue lines represent paddle locations before loadings, green lines represent after loadings. Note that the right paddle is fixed for all loadings.

Comparison of paddle locations in four loadings. Note that the right paddle is fixed for all loadings.

The reason that four sets of loadings are applied in forward problem is to provide more information to reconstruct material parameters. Most of inverse problem in elasticity is nonuniqueness. Previous research [

Given material parameters and loadings, the displacements and forces can be calculated. The surface force will be used as input to reconstruction material parameters in the inverse problem. In fact, surface displacement and force are equivalent as input to solve inverse problem. Most of previously research applied displacement in linear elastography. In this nonlinear elastography study, however, it is found the reconstruction is more sensitive to force than displacement. Therefore surface force is measured and compared with calculated force to reconstruct material parameters.

Reconstruction for nonlinear elastic moduli in 3D breast phantom take input extracted from the deformation in response to loading modes A

For three materials, elastic parameters are:

Initial estimate and reconstructed results for nonlinear elastogrpahy. The results in the first part are based on ideal input (without noise). The ones in the second part are based on input with 5% noise and regularization is not used. The third part is based on input with 5% noise and regularization is applied to reduce the impact of noise. (

Fatty | Glandular | Tumor | |||||||

Real | 35 | 12.5 | 0.4 | 50 | 25 | 0.25 | 80 | 35 | 1.5 |

Guess | 20 | 10 | 1 | 20 | 10 | 1 | 20 | 10 | 1 |

Ideal input | |||||||||

Recon | 34.9988 | 12.5004 | 0.3999 | 50.0512 | 24.9998 | 0.2498 | 81.5331 | 34.9423 | 1.5003 |

5% Noise, without regularization | |||||||||

Recon | 22.2753 | 8.4147 | 1.5038 | 56.1970 | 21.5414 | 0.2509 | 0.0001 | 41.5562 | 2.0010 |

5% Noise, with exponential form regularization | |||||||||

Recon | 37.56290 | 12.48318 | 0.4592 | 50.0014 | 25.0042 | 0.2444 | 80.0002 | 39.1539 | 1.5020 |

Elastography includes forward and inverse problem. In forward problem, material parameters and loadings are given to calculate the deformation; while in inverse problem, external loadings and deformation are known to reconstruct material parameters. Most researchers established certain objective function and minimized it with a proposed iterative algorithm. The challenge is how to calculate the gradient of objective function efficiently and accurately. A straightforward calculation of gradients requires solving stiffness matrix in each iteration, which takes most of the time consumed in the finite element method.

In this study an adjoint method is employed to analytically calculate the gradients. Oberai et al. [

Above results are based on four sets of force measurements on surface. For 2D isotropic elastography, Barbone and Bamber [

The key points for using multiple sets of measurements are to bring more deformation modes simultaneously into consideration. The loadings should be close to tumors in order to make tumors have larger deformation. In Figures

The above results are based on ideal input. However, noise cannot be avoidable in experiments. Its impact on reconstruction is therefore investigated with 5% of noise applied on surface measurements. The results are shown in the second part of Table

Specific forms of penalty term have been designed for different problems [

The results are shown in the third part of Table

This paper presents a study on nonlinear elastography of biomedical imaging, in which a 3D model is developed for heterogeneous breast tissues extracting from real images including fatty tissue, glandular tissue, and tumors. Based on the large-deformation constitutive law, discretized nonlinear equations are solved for displacement, strain, and stress fields in breast tissues with given tumors under external compression at breast boundaries. A 3D inverse-problem algorithm is developed to reconstruct the material parameters for nonlinear elastic constitutive relation of breast phantoms with tumors. The adjoint gradient method is introduced to improve the numerical efficiency and enhance the stability of elastography reconstruction. Results demonstrate that this work opens the door toward nonlinear elastography, and provides guidelines for future developments and clinical applications in breast cancer study.

A classic quasi-Newton iterative method [

It is difficult to calculate gradient

By introducing the adjoint method, it seems that more equations and variables (

This work is supported by the US Army’s Breast Cancer Research Program Concept Award (W81XWH-05-1-0461).