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!