1/20 joyce mclaughlin rensselaer polytechnic institute interior elastodynamics inverse problems i:...
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Joyce McLaughlin
Rensselaer Polytechnic Institute
Interior Elastodynamics Inverse Problems I: Basic Properties
IPAM – September 9, 2003
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Interior Elastodynamics Inverse Problems
Data: Propagating Elastic Wave
• Initially the medium is at rest
• Time and space dependent interior displacement
measurements
• Wave has a propagating front
Characteristics:
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Our Application: Transient Elastography
Goal: Create image of shear wave speed in tissue
• 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
Characteristics of the applicaton:
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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
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5/20Experimental Setup
The device that is used by Fink’s Lab to excite the target tissueand to measure the shear wave motion at the same time
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Two Bar Transducer
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Cross Correlation
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Mathematical Models
OR
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How rich is the data set?
Proof
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Proof (continued)
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Elasticity Case
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13/20How rich is the data set?
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Proof
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Theorems we use
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16/20Equations of Elasticity
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17/20 Nonuniqueness in Anisotropic Media
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Sketch of Proof
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Nonuniqueness Example (The simplest)
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Conclusion1. Data set is rich.2. Data identifies more than one physical property.3. Arrival Time is a particularly rich data set.
1. In the elastic case, how is shear wave front defined and used for identification?
2. What if not all the components of the elastic displacement vector are measured?
3. When additional physical properties are to be determined what are the continuous dependence results?
4. 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?
Open Questions