Linear and Nonlinear Elastic Modulus Imaging: An Application to Breast Cancer Diagnosis

We pose the inverse elasticity problem as a constrained minimization problem as follows. Given n measured displacement fields umeas1,…,umeas,nu1meas,…,unmeas, find the material properties β = [β1, β2] = [μ, γ] such that the objective function is minimized under the constraint that the predicted displacement fields satisfy the equations of equilibrium and boundary conditions given in (3)–(5). In other words, we seek that distribution of material parameters [μ, γ] which leads to the predicted displacement fields ui that attain the global minimum in (1). The second term in (1) is the regularization term and ameliorates the ill-posed nature of the inverse problem. The regularization parameter αj is noise dependent and can be determined using the L-curve method or Morozov’s principle [41]. We have chosen the total variation diminishing (TVD) regularization [42] which is implemented in the form Here, c is chosen to be a small constant to ensure that the regularization contribution is smooth when ∇ βj = 0. The regularization contribution may be thought of as a penalty term that controls the smoothness of the solution. TVD regularization has proven to work well for our purpose as it penalizes fluctuations in the parameters without regard to their steepness and thus preserves sharp inclusion boundaries. Here,Fi is the deformation gradient and Si is the second Piola–Kirchhoff stress tensor, which is derived from the strain energy density function (discussed in the following paragraph). Tractions are denoted by hi, prescribed displacements by gi, and n is the boundary outward unit normal vector. The boundaries for the prescribed displacements on Гg and for the traction on Гh are assumed to satisfy the following conditions: ∂Ω0=Γg∪Γh¯.and Гg ∩ Гh = ∅. We discuss the boundary conditions in more detail below. We have used a modified version of the Veronda–Westmann strain energy density function for an incompressible material in three dimensions. It consists of one linear parameter and one nonlinear parameter as opposed to the original Veronda–West-mann model [43] with three hyperelastic parameters. The strain energy density is given by Here, I1 and I2 are, respectively, the first and second principal invariants of the Cauchy Green tensor, C = FT F. Further μ represents the shear modulus of the material at small strains and γ is a parameter that determines the exponential increase in stiffness with increasing strain. We refer to γ as the nonlinear parameter, or simply, the “nonlinearity.” Under plane stress conditions, the strain energy density function above yields the following expression for the second Piola–Kirchhoff stress tensor Sevan Goenezen, Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180 USA. Jean-Francois Dord, Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180 USA. Zac Sink, Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180 USA. Jingfeng Jiang, Department of Medical Physics, University of Wisconsin-Madison, Wisconsin, WI 53706 USA. Timothy J. Hall, Department of Medical Physics, University of Wisconsin-Madison, Wisconsin, WI 53706 USA. Assad A. Oberai, Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180 USA. We are experimenting with display styles that make it easier to read articles in PMC. Our first effort uses eBook readers, which have several 'ease of reading' features already built in. These PMC articles are best viewed in the iBooks reader. You may notice problems with the display of certain parts of an article in other eReaders. Source.


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