1. 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

2. Magnetic Resonance Elastography, MRE
A newly developed non-destructive technique
(Muthupillai et al., Science, 269, , 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

3. MRE System in Hokkaido Univ. (JST Proj.)
Japan Science and Technology Agency (JST)
GFRP Bar
2~4 m
Micro-MRI
Electromagnetic vibrator
Object
External
vibration
system
(1) External vibration system
(2) Pulse sequence with motion-sensitizing gradients (MSG)
(3) phantom
Wave image
Japan Science and Technology Agency (JST)
Storage modulus
(4) Inversion algorithm

4. MRE phantom: agarose or PAAm gel
100mm
70mm
65mm
10 mm
hard
soft
--- time harmonic external vibration (3D vector)
--- frequency of external vibration (50～250Hz)
--- amplitude of external vibration (≤ 500 μm )

5. Viscoelastic wave in soft tissues
Time harmonic external vibration
Interior viscoelastic wave
　--- amplitude of viscoelastic wave
( : real part, : imaginary part)
viscoelastic body
after some time

6. motion-sensitizing gradients
MRI pulse sequence: SE
MRE pulse sequence: SE+MSG TR TR RF

 RF


slice
slice
Spin echo pulse
sequence
for MRI
MSG
motion-sensitizing gradients
phase
phase
readout
readout
Vibration
echo
signal
echo
signal
with weak wave information
with strong wave information
without wave information

7. MRE measurements: phase image
MRI signal
２D FFT
real part: R
imaginary part: I
magnitude image
phase image
MRE measurement

8. MRE measurements: phase image
磁気回転比
components in vertical direction
(unit: )

9. MRE measurements: phase image
磁気回転比
components in vertical direction
(unit: )

10. viscoelasticity models for soft tissues or phantom
Data analysis for MRE
viscoelasticity of
soft tissues or phantom　　　　　　
interior wave displacement 　　　　　　
Step 3: recovery
（inverse problem）　　　　
Step 2: numerical simulation
（forward problem）　　　　
Step 1: modeling　　　　　
viscoelasticity models for
soft tissues or phantom
(PDE)

11. Viscoelasticity models for soft tissues
Time:
: bounded domain;
: Lipschitz continuous boundary;
Displacement:
General linear viscoelasticity model:

12. Viscoelasticity models for soft tissues
Stress tensor:
Density:
Small deformation (micro meter) ⇒ linear strain tensor

13. Constitutive equation
Voigt model:
Maxwell model:
Zener model:

14. Viscoelasticity tensors
full symmetries:
strong convexity (symmetric matrix ):

15. 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), , 1994.
Rivera, et. al., Comm. Math. Phys. 177(3), 583–602, 1996.

16. 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

17. Constitutive equation (stationary case)
Voigt model:
Maxwell model:
Zener model:

18. 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

19. Storage modulus and loss modulus
Storage ・ loss modulus ( )
Voigt model
Maxwell model
Zener
Angular frequency:
Shear modulus:
Shear viscosity:
Measured by rheometer

20. Modified Stokes model 2D numerical simulation (Freefem++)
Plane strain assumption mm

21. Curl operator: filter of the pressure term
Modified Stokes model:
Constants :
Curl operator: filter of the pressure term

22. 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

23. Denoising

24. Recovery of storage modulus
Constants:
Mollification:
Curl operator:
Numerical differentiation method
Numerical differentiation is an ill-posed problem
Numerical differentiation with Tikhonov regularization
Unstable!!!

25. Recovery of storage modulus
Constants:
Mollification:
Curl operator:
Numerical Integration Method
: test region (2D or 3D)
: test function
Unstable!!!

26. Recovery of storage modulus
Constants:
Mollification:
Curl operator:
Numerical Integration Method
: test region (2D or 3D)
: test function
Unstable!!!

27. Recovery from no noise simulated data
Inclusion: small large outside
Exact value: kPa kPa kPa
Mean value: kPa kPa kPa
Stddev:
Relative error:

28. Recovery from noisy simulated data
10% relative error
Inclusion: small large outside
Exact value: kPa kPa kPa
Mean value: kPa kPa kPa
StdDev:
Relative error:

29. Recovery from experimental data
Layered PAAm gel: hard (left) soft (right)
Mean value: kPa kPa
StdDev:
250 Hz
0.3 mm cm
kPa

30. Recovery of storage modulus G’
Layered PAAm gel: hard (left) soft (right)
Mean value: (25.974) kPa (8.988) kPa
Standard deviation: (6.982) (4.407)
modified method (old method (polynomial test function)) cm
kPa
250 Hz
0.3 mm cm
kPa

31. Recovery of storage modulus G’
250 Hz
0.3 mm
125 Hz
0.5 mm

32. Recovery of storage modulus G’
Layered PAAm gel: hard (left) soft (right)
Mean value: kPa kPa
StdDev:
125 Hz
0.5 mm cm
kPa

33. Recovery of storage modulus G’
hard soft MRE, 250 Hz: kPa kPa MRE, 125 Hz: kPa kPa StdDev, 250 Hz: StdDev, 125 Hz: cm
kPa
250 Hz
0.3 mm
125 Hz
0.5 mm

34. Recovery of storage modulus G’
Wavelength [mm]
○ Resolution of MRE:
0.5 ~ 1 wavelength
○ Best size of object (phantom):
> 2 × wavelength
○ Viscosity: high frequency
 attenuation
○ Best FOV of Micro-MRI:
~ 70 mm × 70 mm × 100 mm
frequency f [Hz]
250
125
62.5
31.25
Storage modulus G’ [kPa] 1
4.0
8.0
16.0
32.0 2
5.7
11.3
22.6
45.3 3
6.9
13.9
27.7
55.4 4
64.0 5
8.9
17.9
35.8
71.6 6
9.8
19.6
39.2
78.4 7
10.6
21.2
42.3
84.7 8
90.5 9
12.0
24.0
48.0
96.0 10
12.6
25.3
50.6
101.2 11
13.3
26.5
53.1
106.1 12
110.9 13
14.4
28.8
57.7
115.4 14
15.0
29.9
59.9
119.7 15
15.5
31.0
62.0
123.9
Wavelength
第３８回　日本磁気共鳴医学会大会　　シンポジウムV　　MR Elastography
平成２２年１０月２日（土）＠つくば国際会議場　第一会場（大ホール　２F）
千葉大学　菅幹生

35. Rheometer Rheometer : ARES-2KFRT, TA Instruments
Frequency: 0.1 ~ 10 Hz
Strain mode: 5%
100mm
65mm
70mm
15 mm
hard
soft
4 ~ 8 mm
1.8 mm

36. Rheometer Rheometer 1: ARES-2KFRT, TA Instruments
Frequency: 0.1 ~ 10 Hz
Strain: 5% and 10%
100mm
65mm
70mm
15 mm
hard
soft
4 ~ 8 mm
1.8 mm

37. Recovery of storage modulus G’
(5% (10%)) hard soft Rheometer 1: ( ) (9.1176) Rheometer 2: ( ) (5.3534)
Independent of frequencies
(0.1 ~ 10 Hz)

38. Recovery of storage modulus G’
Independent of frequencies (1 ~ 250 Hz)
(5% (10%)) hard soft Rheometer 1: ( ) (9.1176) Rheometer 2: ( ) (5.3534) MRE, 250 Hz: kPa kPa Relative error 1: (0.0528) (0.1804) Relative error 2: (0.0389) (1.0103) > 2 waves > 4 waves MRE, 125 Hz: kPa kPa Relative error 1: (0.7878) (0.1697) Relative error 2: (0.8063) (0.9922) only have 1 wave > 2 waves

39. Recovery of storage modulus G’
Independent of frequencies (1 ~ 250 Hz)
hard soft Rheometer: kPa kPa MRE, 250 Hz: kPa kPa Relative error:
100mm
65mm
70mm
15 mm
hard
soft
4 ~ 8 mm
Rheometer : ARES-2KFRT, TA Instruments
Frequency: 0.1 ~ 10 Hz
Strain mode: 5%

40. Recovery of storage modulus G’
Independent of frequencies (1 ~ 250 Hz)
hard soft Rheometer: kPa kPa MRE, 250 Hz: kPa kPa Relative error:
Rheometer : ARES-2KFRT, TA Instruments
Frequency: 0.1 ~ 10 Hz
Strain mode: 5%

41. Hölder stability Scalar model: Viscoelasticity:
Able to show the Hölder stability for analytic complex coefficients case (n=2 or 3):
Alessandrini, Anna. di Mate. Pura ed Appl., (2D, elliptic, real coefficient)
Lin et. al., (doubling inequalities, three-sphere inequalities)

42. Conclusions For noisy simulated data with 10% relative error:
relative error of recovery is good
(less than or about 15%);
For experimental data, if we have more than 2 waves inside, the recovery is good;

43. Thank you for your attentions!

44. MRE for human liver MRE Huwart 1 Yin 2 Asbach 3 Motosugi Country
France
USA
Germany
Japan
Year
2007
2010 n 88
35+50 85
>F1
2.4
2.93
2.9
2.6
AUC
0.96
0.99
0.92
0.94
>F2
2.5
4.89
3.2
3.5
.99
0.95
>F3
3.1
6.47
4.0
4.1
0.98
>F4
4.3
5.5
4.4
1.00
Huwart L, et al. Radiology 2007; 245:458-66
Yin M, et al. Clin Gastroenterol Hepatol 2007; 5:
Asbach, et al. Radiology 2010 online
Motosugi, et al.
Univ. of Yamanashi