part 4: viscoelastic properties of soft tissues in a living body measured by mr elastography
DESCRIPTION
Part 4: Viscoelastic Properties of Soft Tissues in a Living Body Measured by MR Elastography Gen Nakamura Department of Mathematics, Hokkaido University, Japan (Supported by Japan Science and Technology Agency) - PowerPoint PPT PresentationTRANSCRIPT
Part 4: Viscoelastic Properties of Soft Tissues in a Living Body Measured by MR Elastography
Gen Nakamura Department of Mathematics, Hokkaido University, Japan (Supported by Japan Science and Technology Agency)
Joint work with Yu, Jiang
ICMAT, Madrid, May 12, 2011
Magnetic Resonance Elastography, MRE
A newly developed non-destructive technique (Muthupillai et al., Science, 269, 1854-1857, 1995, Mayo Clinic.)
Measure the viscoelasticity of soft tissues in a living body Diagnosis:
the stage of liver fibrosis early stage cancer: breast cancer, pancreatic
cancer, prostate cancer, etc. neurological diseases: Alzheimer’s disease,
hydrocephalus, multiple sclerosis, etc. Nondestructive testing (high frequency
rheometer): biological material, polymer material
MRE System in Hokkaido Univ. (JST Proj.)
(1) External vibration system
GFRP Bar 2~4 m
Micro-MRI
Electromagnetic vibrator
ObjectExternal vibration system
(2) Pulse sequence with motion-sensitizing gradients (MSG)
Storage modulus
(4) Inversion algorithm
Wave image
(3) phantom
Japan Science and Technology Agency (JST)
MRE phantom: agarose or PAAm gel
100mm
70mm
65mm
10 mm
--- time harmonic external vibration (3D vector) --- frequency of external vibration (50 ~ 250Hz) --- amplitude of external vibration (≤ 500 μm )
hard soft
Viscoelastic wave in soft tissues Time harmonic external vibration
Interior viscoelastic wave
--- amplitude of viscoelastic wave
( : real part, : imaginary part)
viscoelastic bodyafter some time
MRE measurements: phase imageMRI signal
2 D FFTreal part: R imaginary part: I
magnitude image
phase image
MRE measurement
MRE measurements: phase image
components in vertical direction
(unit: )
Data analysis for MREviscoelasticity of
soft tissues or phantom
viscoelasticity models for soft tissues or phantom
(PDE)
interior wave displacement
Step 1: modeling
Step 2: numerical simulation
( forward problem )
Step 3: recovery( inverse
problem )
Viscoelasticity models for soft tissues Time: : bounded domain; : Lipschitz continuous boundary; Displacement: General linear viscoelasticity model:
Viscoelasticity models for soft tissues Stress tensor: Density:
Small deformation (micro meter) ⇒ linear strain tensor
Constitutive equation Voigt model:
Maxwell model:
Zener model:
Viscoelasticity tensors full symmetries:
strong convexity (symmetric matrix ):
Time harmonic wave Boundary: : open subsets of with , Lipschitz
continuous; Time harmonic boundary input and initial condition:
Time harmonic wave (exponential decay):
Jiang, et. Al., submitted to SIAM appl. math.. (isotropic, Voigt) Rivera, Quar. Appl. Math., 3(4), 629-648, 1994. Rivera, et. al., Comm. Math. Phys. 177(3), 583–602, 1996.
Time harmonic wave Stationary model:
Sobolev spaces of fractional order 1/2 or 3/2 an open subset with a boundary away from and the set of distributions in the usual fractional
Sobolev space compactly supported in This can be naturally imbedded into
Constitutive equation (stationary case) Voigt model:
Maxwell model:
Zener model:
Modified Stokes model Isotropic+ nearly incompressible Asymptotic analysis ⇒ modified Stokes model:
Jiang et. al., Asymptotic analysis for MRE, submitted to SIAP
H. Ammari, Quar. Appl. Math., 2008: isotopic constant elasticity
Storage modulus and loss modulus Storage ・ loss modulus ( )
Voigt model
Maxwell model
Zener
Angular frequency: Shear modulus: Shear viscosity: Measured by rheometer
Modified Stokes model 2D numerical simulation (Freefem++)
Plane strain assumption
mm
Curl operator Modified Stokes model:
Constants :Curl operator: filter of the pressure term
Pre-treatment: denoising Mollifier (Murio, D. A.: Mollification and Space
Marching)
Smooth function defined in a nbd of : a bounded domain : an extension of to Function : a nonnegative function over such
that and
Denoising
Recovery of storage modulus Constants: Mollification: Curl operator: Numerical differentiation method
Numerical differentiation is an ill-posed problem Numerical differentiation with Tikhonov regularization
Unstable!!!
Recovery of storage modulus Constants: Mollification: Curl operator: Numerical Integration Method
: test region (2D or 3D) : test function
Unstable!!!
Recovery of storage modulus Constants: Mollification: Curl operator: Numerical Integration Method
: test region (2D or 3D) : test function
Unstable!!!
Recovery from no noise simulated data
Inclusion: small large outside Exact value: 3.3 kPa 3.3 kPa 7.4 kPa Mean value: 3.787 kPa 3.768 kPa 7.436 kPa Stddev: 0.147 0.060 0.003 Relative error: 0.1476 0.1418 0.00049
Recovery from noisy simulated data
10% relative errorInclusion: small large outside
Exact value: 3.3 kPa 3.3 kPa 7.4 kPa Mean value: 4.636 kPa 3.890 kPa 7.422 kPa StdDev: 0.328 0.129 0.322 Relative error: 0.4048 0.1788 0.00294
Layered PAAm gel: hard (left) soft (right)
Mean value: 31.100 kPa 10.762 kPa
StdDev: 0.535 0.201
250 Hz0.3 mm
cm kPa
Recovery from experimental data
Layered PAAm gel: hard (left) soft (right)
Mean value: 31.100 (25.974) kPa 10.762 (8.988) kPa
Standard deviation: 0.535 (6.982) 0.201 (4.407)
modified method (old method (polynomial test function))
Recovery of storage modulus G’
cm kPa
cm kPa
250 Hz0.3 mm
Recovery of storage modulus G’
hard soft
Rheometer: 32.5456 kPa 9.2472 kPa
MRE, 250 Hz: 31.100 kPa 10.762 kPa
Relative error: 0.0444 0.1638
Independent of frequencies (1 ~ 250 Hz)
Rheometer : ARES-2KFRT, TA InstrumentsFrequency: 0.1 ~ 10 HzStrain mode: 5%
Thank you for your attentions!