Fractional calculus & statistic modeling of anomalous acoustic dissipation with applications to medical ultrasonics

 

( Wen Chen's WWW for acoustics in imaging and exploration)



Key words: imaging, frequency dependent, attenuation, acoustics, anomalous diffusion, fractional calculus, fractional Laplacian, Caputo, Riemann-Liouville, Riesz potential, Lévy distribution, fractal, wave, vibration, electromagnetic wave, damping, ultrasound, power law.

 

The research described below is a part of the project "mathematical and numerical modelings of medical ultrasound wave propagation", for which I was the project manager (other team memebers: Aicha Bounaim, Xing Cai, Sverre Holm, Aslak Tveito, Åsmund Ødegård). This project is of multidisciplinary undertaking, involving acoustics, medical imaging, signal processing, mechanics, statistic physics, partial differential equation modeling, scientific computing, etc. Besides the earlier work in the simulation of beam forming through linear and nonlinear media related to transducers designs, the two major subtopics of the project are

1) simulation of clinical amplitude/velocity reconstruction imaging technique for breast tumor and ultrasonic detection of bone density (see 2D & 3D simulation results)

2) anomalous attenuation (dissipation) of ultrasound wave propagations, which plays a prominent role in many medical ultrasound applications, for instance, the ultrasound second harmonic imaging and the high intensity focused ultrasound beam for hyperthermic surgery.

Below is a list of our relevant publications regarding this study:

·   W. Chen, A speculative study of 2/3-order fractional Laplacian modeling of turbulence: Some thoughts and conjectures, Chaos, 16, 023126, 2006.

·   W. Chen, Time-space fabric underlying anomalous diffusion, Soliton, Fractal, & Chaos, 28(4), 923-929, 2006.

·   W. Chen, L¨¦vy stable distribution and [0,2] power law dependence of acoustic absorption on frequency in various lossy media, Chinese Physics Letter(ÖйúÎïÀí¿ì±¨)£¬22(10)£¬2601-2603, 2005.

·   W. Chen, S. Holm, A. Bounaim, A. Odegard and A. Tveito, A Frequency Decomposition Time Domain Model of Broadband Frequency-Dependent Absorption, Int. J. of Tomography and Statistics, 2(4), 15-26, 2005.

·   W. Chen and S. Holm, Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency dependency, J. Acoustic Society of America, 115(4), 1424-1430, 2004.

·   A. Bounaim, S. Holm, W. Chen, A. Odegard, A. Tveito and K. Thomenius, Sensitivity of the ultrasonic CARI technique for breast tumor detection using a FETD scheme, Ultrasonics,42, 919-925, 2004.

·   A. Bounam, S. Holm, W. Chen, and A. Odegard, Quantification of the CARI breast imaging sensitivity by 2D/3D numerical time-domain ultrasound wave propagation,Mathematics and Computers in Simulation, 65, 521¨C534£¬2004.

·   W. Chen and S. Holm, Modified Szaboo wave equation models for lossy media obeying frequency power law, J. Acoustic Society of America, 2570-2574, 114(5), 2003.

This site mainly focuses on the second part of this project. Among our major contributions in this regard are

the new definition of the fractional Laplacian, aslo known as Riesz fractional derivative, which overcomes the hyper-singularity of the traditional definition and naturallly includes boundary conditions in finite domains;

the development of the linear and nonlinear causal fractional Laplacian wave equations and the corresponding FEM numerical models for lossy media exhibiting arbitrary frequency power law attenuation, with the classical proportional Rayleigh damping being a special case;

the first mathematical physics explanation of [0,2] power dependency of attenuation coefficient on frequency via the Levy stable distribution theory;

the introduction of the concept of the positive fractional time derivative and accordingly the presentation of the modified Szabo wave equations, where the hyper-singularity of the original Szabo wave equation models for anomalously attenuative media is eased and the integer-order initial condition is naturally included;

the establishment of explicit links between fractional calculus equation models, 1/f power spectrum, Hurst exponent, fractals, fractional Brownian motion and Levy stable process, all of which reflect the memory dynamics and/or fractal (topology/molecular structure) microstructures of complex systems.

These works can be easily extended to other acoustic applications such as geophysical explorations and nondestructive detections. Here is a recent internal project seminar. For more details, please also go to a summary and the related publications.


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