27,219 research outputs found
Viscoelastic modulus reconstruction using time harmonic vibrations
This paper presents a new iterative reconstruction method to provide
high-resolution images of shear modulus and viscosity via the internal
measurement of displacement fields in tissues. To solve the inverse problem, we
compute the Fr\'echet derivatives of the least-squares discrepancy functional
with respect to the shear modulus and shear viscosity. The proposed iterative
reconstruction method using this Fr\'echet derivative does not require any
differentiation of the displacement data for the full isotropic linearly
viscoelastic model, whereas the standard reconstruction methods require at
least double differentiation. Because the minimization problem is ill-posed and
highly nonlinear, this adjoint-based optimization method needs a very
well-matched initial guess. We find a good initial guess. For a well-matched
initial guess, numerical experiments show that the proposed method considerably
improves the quality of the reconstructed viscoelastic images.Comment: 15 page
Lam\'e Parameter Estimation from Static Displacement Field Measurements in the Framework of Nonlinear Inverse Problems
We consider a problem of quantitative static elastography, the estimation of
the Lam\'e parameters from internal displacement field data. This problem is
formulated as a nonlinear operator equation. To solve this equation, we
investigate the Landweber iteration both analytically and numerically. The main
result of this paper is the verification of a nonlinearity condition in an
infinite dimensional Hilbert space context. This condition guarantees
convergence of iterative regularization methods. Furthermore, numerical
examples for recovery of the Lam\'e parameters from displacement data
simulating a static elastography experiment are presented.Comment: 29 page
Constitutive Models for Tumour Classification
The aim of this paper is to formulate new mathematical models that will be able to differentiate not only between normal and abnormal tissues, but, more importantly, between benign and malignant tumours. We present preliminary results of a tri-phasic model and numerical simulations of the effect of cellular adhesion forces on the mechanical properties of biological tissues.
We pursued the following three approaches:
(i) the simulation of the time-harmonic linear elastic models to examine coarse scale effects and adhesion properties,
(ii) the investigation of a tri-phasic model, with the intent of upscaling this model to determine effects of electro-mechanical coupling between cells,
and (iii) the upscaling of a simple cell model as a framework for studying interface conditions at malignant cells.
Each of these approaches has opened exciting new directions of research that we plan to study in the future
Progress on the Strong Eshelby's Conjecture and Extremal Structures for the Elastic Moment Tensor
We make progress towards proving the strong Eshelby's conjecture in three
dimensions. We prove that if for a single nonzero uniform loading the strain
inside inclusion is constant and further the eigenvalues of this strain are
either all the same or all distinct, then the inclusion must be of ellipsoidal
shape. As a consequence, we show that for two linearly independent loadings the
strains inside the inclusions are uniform, then the inclusion must be of
ellipsoidal shape. We then use this result to address a problem of determining
the shape of an inclusion when the elastic moment tensor (elastic
polarizability tensor) is extremal. We show that the shape of inclusions, for
which the lower Hashin-Shtrikman bound either on the bulk part or on the shear
part of the elastic moment tensor is attained, is an ellipse in two dimensions
and an ellipsoid in three dimensions
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