Fractal ladder models and power law wave equations

Abstract

The ultrasonic attenuation coefficient in mammalian tissue is approximated by a frequency-dependent power law for frequencies less than 100 MHz. To describe this power law behavior in soft tissue, a hierarchical fractal network model is proposed. The viscoelastic and self-similar properties of tissue are captured by a constitutive equation based on a lumped parameter infinite-ladder topology involving alternating springs and dashpots. In the low-frequency limit, this ladder network yields a stress-strain constitutive equation with a time-fractional derivative. By combining this constitutive equation with linearized conservation principles and an adiabatic equation of state, a fractional partial differential equation that describes power law attenuation is derived. The resulting attenuation coefficient is a power law with exponent ranging between 1 and 2, while the phase velocity is in agreement with the Kramers-Kronig relations. The fractal ladder model is compared to published attenuation coefficient data, thus providing equivalent lumped parameters.

Generalized viscoelastic wave equation

“…A fundamental characteristic of the viscoelastic media is that the attenuation satisfies power laws of frequency, for both compressional and shear waves (Collins & Lee 1956;White 1965;Kibblewhite 1989;Szabo & Wu 2000;GómezÁlvarez-Arenas et al 2002;Fabry et al 2003;Nicolle et al 2005;Kelly & McGough 2009;Treeby et al 2010). In this section, we will demonstrate that the generalized wave equation has a solution of attenuation that satisfies the power laws and is consistent with many laboratory measurements.…”

Optimization of a label‐free biosensor vertically characterized based on a periodic lattice of high aspect ratio SU‐8 nano‐pillars with a simplified 2D theoretical model

“…Moreover, for some individual examples the local value of γ exhibit lower values at low ultrasound frequencies and tends to 2 in the high frequency limit . Despite the study of the physical mechanism besides this complex frequency dependence is out of the scope of this work, there exist numerous phenomenological approaches for including the observed losses in the acoustic equations (Wismer et al, 1995;Kellya et al, 2009). On the other hand, is common in literature to describe the losses in soft tissues as multiple-relaxation processes (Nachman et al, 1990;, from continuous distribution of relaxation processes (Jongen et al, 1986) to a discrete representation.…”

Approximation of potential-driven flow dynamics in large-scale self-similar tree networks

“…Nakagawa & Sorimachi (1992) do this in the case of an infinite resistor-capacitor circuit with a physically repeated pattern to create a simpler, finite macromodel circuit, allowing the dynamic solution of a previously intractable mathematical problem. Similar approaches have been used with other infinite electrical networks, including networks with differing forms of similarity such as ladders, grids and rings (Zemanian 1991;Srinivasan 1992;Mavromatis 1995;Thompson 1997;Kelly & McGough 2009). In each of these cases, models are reduced by taking advantage of physical structure and the infinite nature of the system.…”