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.) 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

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 )

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

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

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

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

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

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)

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: filter of the pressure term 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 error Inclusion: 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

Recovery from experimental data Layered PAAm gel: hard (left) soft (right) Mean value: 31.100 kPa 10.762 kPa StdDev: 0.535 0.201 250 Hz 0.3 mm cm kPa

Recovery of storage modulus G’ 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)) cm kPa 250 Hz 0.3 mm cm kPa

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

Recovery of storage modulus G’ Layered PAAm gel: hard (left) soft (right) Mean value: 6.269 kPa 10.665 kPa StdDev: 0.579 0.134 125 Hz 0.5 mm cm kPa

Recovery of storage modulus G’ hard soft MRE, 250 Hz: 31.100 kPa 10.762 kPa MRE, 125 Hz: 6.269 kPa 10.665 kPa StdDev, 250 Hz: 0.535 0.201 StdDev, 125 Hz: 0.579 0.134 cm kPa 250 Hz 0.3 mm 125 Hz 0.5 mm

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

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

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

Recovery of storage modulus G’ (5% (10%)) hard soft Rheometer 1: 32.5456 (29.5410) 9.2472 (9.1176) Rheometer 2: 29.2902 (32.3578) 5.9797 (5.3534) Independent of frequencies (0.1 ~ 10 Hz)

Recovery of storage modulus G’ Independent of frequencies (1 ~ 250 Hz) (5% (10%)) hard soft Rheometer 1: 32.5456 (29.5410) 9.2472 (9.1176) Rheometer 2: 29.2902 (32.3578) 5.9797 (5.3534) MRE, 250 Hz: 31.100 kPa 10.762 kPa Relative error 1: 0.0444 (0.0528) 0.1638 (0.1804) Relative error 2: 0.0618 (0.0389) 0.7998 (1.0103) > 2 waves > 4 waves MRE, 125 Hz: 6.269 kPa 10.665 kPa Relative error 1: 0.8074 (0.7878) 0.1533 (0.1697) Relative error 2: 0.7860 (0.8063) 0.7835 (0.9922) only have 1 wave > 2 waves

Recovery of storage modulus G’ Independent of frequencies (1 ~ 250 Hz) hard soft Rheometer: 32.5456 kPa 9.2472 kPa MRE, 250 Hz: 31.100 kPa 10.762 kPa Relative error: 0.0444 0.1638 100mm 65mm 70mm 15 mm hard soft 4 ~ 8 mm Rheometer : ARES-2KFRT, TA Instruments Frequency: 0.1 ~ 10 Hz Strain mode: 5%

Recovery of storage modulus G’ Independent of frequencies (1 ~ 250 Hz) hard soft Rheometer: 32.5456 kPa 9.2472 kPa MRE, 250 Hz: 31.100 kPa 10.762 kPa Relative error: 0.0444 0.1638 Rheometer : ARES-2KFRT, TA Instruments Frequency: 0.1 ~ 10 Hz Strain mode: 5%

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., 1986. (2D, elliptic, real coefficient) Lin et. al., http://arxiv.org/abs/0802.1983v1, (doubling inequalities, three-sphere inequalities)

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;

Thank you for your attentions!

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: 1207-13 Asbach, et al. Radiology 2010 online Motosugi, et al. Univ. of Yamanashi