Abstract
The identifiability of the two damping components of a Generalized Rayleigh Damping model is investigated through analysis of the continuum equilibrium equations as well as a simple spring-mass system. Generalized Rayleigh Damping provides a more diversified attenuation model than pure Viscoelasticity, with two parameters to describe attenuation effects and account for the complex damping behavior found in biological tissue. For heterogeneous Rayleigh Damped materials, there is no equivalent Viscoelastic system to describe the observed motions. For homogeneous systems, the inverse problem to determine the two Rayleigh Damping components is seen to be uniquely posed, in the sense that the inverse matrix for parameter identification is full rank, with certain conditions: when either multi-frequency data is available or when both shear and dilatational wave propagation is taken into account. For the multi-frequency case, the frequency dependency of the elastic parameters adds a level of complexity to the reconstruction problem that must be addressed for reasonable solutions. For the dilatational wave case, the accuracy of compressional wave measurement in fluid saturated soft tissues becomes an issue for qualitative parameter identification. These issues can be addressed with reasonable assumptions on the negligible damping levels of dilatational waves in soft tissue. In general, the parameters of a Generalized Rayleigh Damping model are identifiable for the elastography inverse problem, although with more complex conditions than the simpler Viscoelastic damping model. The value of this approach is the additional structural information provided by the Generalized Rayleigh Damping model, which can be linked to tissue composition as well as rheological interpretations.
Highlights
The importance of damping models in elastography has become clearer in recent years as attenuation levels measured by elastographic imaging have been linked to diseases of the liver [1,2,3,4,5,6] and brain [7,8,9,10,11,12,13]
The goal of this paper is to investigate the conditions in which the parameters for a Generalized Rayleigh Damping (RD) model can be identified in the elastography problem, based on measured motion data within the specimen
A Spring-Mass Analogy To explore the concept of parameter identification in Generalized RD elastography more closely, we start by considering a simple, locally homogeneous spring-mass system, where the spring stiffness, k, has a VE component, k0, while the mass, m, has an inertial damping component, m0
Summary
The importance of damping models in elastography has become clearer in recent years as attenuation levels measured by elastographic imaging have been linked to diseases of the liver [1,2,3,4,5,6] and brain [7,8,9,10,11,12,13]. A number of methods have been proposed for reconstructing the Viscoelastic (VE) properties of soft tissue [14,15,16,17,18], including an iterative, nonlinear inversion method [19,20] These methods have all targeted the development of images of the storage (G0 ) and loss (G00 ) modulus distributions within the tissue in question. Some have gone on to investigate the frequency dependent behavior of these two parameters [5,11], as well as multi-frequency reconstruction methods to improve the quality of the VE parameters reconstruction across a range of frequencies [18] While these methods have already demonstrated the important role of tissue attenuation in differentiating tissue type and identifying lesions, linear VE provides a relatively simplified model for understanding the complex, non-linear attenuation observed in in-vivo tissue. A rheological interpretation of RD can be developed for weak to moderate damping levels [22]
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