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Exploring Interior Elastodynamics: Inverse Problems and Transient Elastography Applications

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This document discusses the basic properties of interior elastodynamics and the challenges of inverse problems. It focuses on the application of transient elastography to generate images of shear wave speed in tissue. The characterization of propagating elastic waves is outlined, emphasizing the importance of measuring interior displacement through ultrasound or MRI. Key points include unique insights into shear wave speeds in abnormal and normal tissues, experimental setups, and the complexities of data interpretation in elastodynamics. The text raises crucial open questions regarding shear wave identification and physical property determination.

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Exploring Interior Elastodynamics: Inverse Problems and Transient Elastography Applications

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  1. Interior Elastodynamics Inverse Problems I: Basic Properties Joyce McLaughlin Rensselaer Polytechnic Institute IPAM – September 9, 2003

  2. Interior Elastodynamics Inverse Problems Data: Propagating Elastic Wave Characteristics: • Initially the medium is at rest • Time and space dependent interior displacement • measurements • Wave has a propagating front

  3. Our Application: Transient Elastography Goal: Create image of shear wave speed in tissue Characteristics of the applicaton: • Shear wavespeed increases 2-4 times in abnormal tissue • Shear wavespeed is 1-3 m/sec in normal tissue • Interior displacement of wave can be measured with ultrasound (or MRI) • Ultrasound utilizes compression wave whose speed is 1500 m/sec • Low amplitude  linear equation model • Wave is initiated by impulse on the boundary  data has central frequency • Data supplied by Mathias Fink

  4. Compare with • Static experiment: tissue is compressed (Ophir) • Dynamic sinusoidal excitation: • time harmonic boundary source (Parker) • Our application: Transient elastography Group members: Lin Ji, Kui Lin, Antoinette Maniatty, Eunyoung Park, Dan Renzi, Jeong-Rock Yoon

  5. Experimental Setup The device that is used by Fink’s Lab to excite the target tissue and to measure the shear wave motion at the same time

  6. Two Bar Transducer

  7. Cross Correlation

  8. Mathematical Models OR

  9. How rich is the data set? Proof

  10. Proof (continued)

  11. Elasticity Case

  12. How rich is the data set?

  13. Proof

  14. Theorems we use

  15. Equations of Elasticity   

  16. Nonuniqueness in Anisotropic Media

  17. Sketch of Proof

  18. Nonuniqueness Example (The simplest)

  19. Conclusion • Data set is rich. • Data identifies more than one physical property. • Arrival Time is a particularly rich data set. Open Questions • In the elastic case, how is shear wave front defined and used for identification? • What if not all the components of the elastic displacement vector are measured? • When additional physical properties are to be determined what are the continuous dependence results? • When there is a large discrepancy in wavespeeds an incompressible model may be appropriate. What are the uniqueness and continuous dependence results in this case?

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