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Part 4: Viscoelastic Properties of Soft
Tissues in a Living Body Measured by MR
Elastography
Gen Nakamura
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Department of Mathematics, Hokkaido University, Japan
(Supported by Japan Science and Technology Agency)
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Joint work with Yu, Jiang
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ICMAT, Madrid, May 12, 2011
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Magnetic Resonance Elastography, MRE
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A newly developed non-destructive technique
(Muthupillai et al., Science, 269, 1854-1857, 1995, Mayo Clinic.)
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Measure the viscoelasticity of soft tissues in a living
body
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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.)
Japan Science and Technology Agency (JST)
Electromagnetic vibrator
GFRP Bar
2~4 m
External
vibration
system
Object
Micro-MRI
(1) External vibration system
(2) Pulse sequence with motion-sensitizing gradients (MSG)
(3) phantom
Wave image
Storage modulus
(4) Inversion algorithm
MRE phantom: agarose or PAAm gel
100mm
65mm
10 mm
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hard
soft
70mm
--- time harmonic external vibration (3D vector)
--- frequency of external vibration (50～250Hz)
--- amplitude of external vibration (≤ 500 μm )
Viscoelastic wave in soft tissues
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Time harmonic external vibration
viscoelastic body
after some time
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Interior viscoelastic wave
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--- amplitude of viscoelastic wave
( : real part, : imaginary part)
MRE measurements: phase image
MRI signal
２D FFT
real part: R
magnitude image
imaginary part: I
phase image
MRE measurement
MRE measurements: phase image
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components in vertical direction
(unit:
)
Data analysis for MRE
viscoelasticity of
soft tissues or phantom
interior wave
displacement
Step 1: modeling
Step 3: recovery
（inverse problem）
Step 2: numerical simulation
（forward problem）
viscoelasticity models for
soft tissues or phantom
(PDE)
Viscoelasticity models for soft tissues
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Time:
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: bounded domain;
: Lipschitz continuous boundary;
Displacement:
General linear viscoelasticity model:
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Viscoelasticity models for soft tissues
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Stress tensor:
Density:
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Small deformation (micro meter) ⇒ linear strain tensor
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Constitutive equation
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Voigt model:
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Maxwell model:
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Zener model:
Viscoelasticity tensors
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full symmetries:
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strong convexity (symmetric matrix
):
Time harmonic wave
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Boundary:
: open subsets of with
, Lipschitz continuous;
Time harmonic boundary input and initial condition:
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Time harmonic wave (exponential decay):
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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
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Stationary model:
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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)
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Voigt model:
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Maxwell model:
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Zener model:
Modified Stokes model
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Isotropic+ nearly incompressible
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Asymptotic analysis ⇒ modified Stokes model:
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Jiang et. al., Asymptotic analysis for MRE, submitted to
SIAP
H. Ammari et. al., Quar. Appl. Math., 2008: isotopic
constant elasticity
Storage modulus and loss modulus
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Storage ・ loss modulus (
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Voigt model
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Maxwell model
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Zener
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Angular frequency:
Shear modulus:
Shear viscosity:
Measured by rheometer
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)
Modified Stokes model
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2D numerical simulation (Freefem++)
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Plane strain assumption
mm
Curl operator
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Modified Stokes model:
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Constants
:
Curl operator: filter of the pressure term
Pre-treatment: denoising
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Mollifier (Murio, D. A.: Mollification and Space Marching)
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Smooth function defined in the nbd of
: a bounded domain
: an extension of to
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Function : a nonnegative
and
function over
such that
Denoising
Recovery of storage modulus
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Constants:
Mollification:
Curl operator:
Unstable!!!
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Numerical differentiation method
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Numerical differentiation is an ill-posed problem
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Numerical differentiation with Tikhonov regularization
Recovery of storage modulus
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Constants:
Mollification:
Curl operator:
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Numerical Integration Method
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: test region (2D or 3D)
: test function
Unstable!!!
Recovery of storage modulus
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Constants:
Mollification:
Curl operator:
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Modified Integral Method
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: test region (2D or 3D)
test region size: about one wavelength
Recovery from no noise simulated data
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Inclusion:
Exact value:
Mean value:
Stddev:
Relative error:
small
3.3 kPa
3.787 kPa
0.147
0.1476
large
3.3 kPa
3.768 kPa
0.060
0.1418
outside
7.4 kPa
7.436 kPa
0.003
0.00049
Recovery from noisy simulated data
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10% relative error
Inclusion:
small
Exact value:
3.3 kPa
Mean value: 4.636 kPa
StdDev:
0.328
Relative error: 0.4048
large
outside
3.3 kPa
7.4 kPa
3.890 kPa 7.422 kPa
0.129
0.322
0.1788
0.00294
Recovery from experimental data
250 Hz
0.3 mm
kPa
cm
Layered PAAm gel:
Mean value:
StdDev:
hard (left)
31.100 kPa
0.535
soft (right)
10.762 kPa
0.201
Recovery of storage modulus G’
cm
kPa
250 Hz
0.3 mm
cm
kPa
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’
Independent of frequencies (1 ~ 250 Hz)
Rheometer:
MRE, 250 Hz:
Relative error:
hard
32.5456 kPa
31.100 kPa
0.0444
Rheometer : ARES-2KFRT, TA Instruments
Frequency: 0.1 ~ 10 Hz
Strain mode: 5%
soft
9.2472 kPa
10.762 kPa
0.1638
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Thank you for your attentions!