1. MR Elastography John B. Weaver
Three essential elements:
Tissue vibration: piezoelectric or remote activation
MRI measurements of tissue motion: phase contrast
Mechanical properties reconstruction: viscoelastic
Dartmouth Contributions:
Intrinsic activation
Finite element material property reconstructions
Poroelastic models
No fibrosis Advanced fibrosis
Kim et al., Radiology 2013

2. Poroelastic Mechanical model at low frequencies
Stress equation
𝛻⋅𝜇 𝛻𝑢+𝛻 𝑢 𝑇 +𝛻 𝜇𝜈 1−2𝜈 𝛻⋅𝑢− 1−𝛽 𝛻𝑃=−(𝜌−𝛽 𝜌 𝑓 ) 𝜔 2 𝑢+𝐹
𝛽= 𝜔 𝜙 2 𝜌 𝑓 𝜅 𝜔𝜅 𝜌 𝑎 +𝜙 𝜌 𝑓 +𝑖 𝜙 2
Pressure equation
𝛻⋅ 𝛽𝛻𝑃 + 𝜌 𝑓 𝜔 2 𝛻 1−𝛽 𝑢 =0
At low frequencies, 𝜔 2 ≈0
𝛻⋅𝜇 𝛻𝑢+𝛻 𝑢 𝑇 +𝛻 𝜇𝜈 1−2𝜈 𝛻⋅𝑢− 1−𝛽 𝛻𝑃=0
Biphasic: Fluid saturated porous matrix
Elastic properties of matrix
Hydrodynamic properties
𝛻⋅ 𝛽𝛻𝑃 =0
Take out a factor of 𝜇 𝑜 , so that 𝜇 𝑥 = 𝜇 𝑜 𝜇 ∗ (𝑥)
Poroelastic
𝜇 𝑜 𝛻⋅ 𝜇 ∗ 𝛻𝑢+𝛻 𝑢 𝑇 +𝛻 𝜇 ∗ 𝜈 1−2𝜈 𝛻⋅𝑢 − 1−𝛽 𝛻𝑃=0
The extra terms in poroelasticity avoid the non-uniqueness issue until very low frequencies where pore fluid pressure equilibrium is reached, 𝛻𝑃→0.
Perrinez PR, Kennedy FE, Van Houten EEW, Weaver JB, Paulsen KD. Modeling of Soft Poroelastic Tissue in Time-Harmonic MR Elastography. IEEE Transactions on Biomedical Engineering 2009, 56(3): 598–608.

3. Non-Linear Inversion Objective Function
Subzone iteration loop
Objective Function
Current Property estimate, qk
Update 𝜃 𝑧𝑜𝑛𝑒 to minimize Φ
𝑈 𝑐 ( 𝜃 𝑘 )=FEM Solution
Forward Problem: Calculate 𝑈 𝑐 𝜃 𝑘 from mechanical model assumption.
Inverse Problem: Estimate 𝜃 𝑘 given the 𝑈 𝑐 displacements that minimize the objective function Φ 𝜃
-Use of iterative algorithms:
Gauss Newton
Conjugate Gradient
Quasi-Newton
kth iteration
pk: search direction
αk: step size
1) Van Houten EE, Paulsen KD, Miga MI, Kennedy FE, Weaver JB. An Overlapping Subzone Technique for MR-Based Elastic Property Reconstruction. Magnetic Resonance in Medicine 1999, 42: 779–786.
2) Van Houten EEW, Miga MI, Weaver JB, Kennedy FE, Paulsen KD. Three-Dimensional Subzone-Based Reconstruction Algorithm for MR Elastography. Magnetic Resonance in Medicine 2001, 45: 827– 837.

4. Build Global Properties
Subzone iteration loop
Non-Linear Inversion
Current Property estimate, qk
Update 𝜃 𝑧𝑜𝑛𝑒 to minimize Φ
𝑈 𝑐 ( 𝜃 𝑘 )=FEM Solution
Objective Function
3D Motion Data
Repeat until property estimate stabilizes
Head Foot
Anterior Posterior
Left Right
3D Finite Element Model
Break into grid of 3D overlapping subzones
Build Global Properties
Union of subzone properties …
Gaussian Smoothing (Spatial Filtering)
Heterogenous Property Description, 𝜃

5. Intrinsic Actuation: IA-MRE (1Hz)
~15% of cardiac output goes to the brain
Carotid pulse pressure is ~6kPa.
MRI can measure the 1Hz motion fields in human brain due to intrinsic pulsation
𝑥 displacement
𝑦 displacement
𝑧 displacement
Weaver JB, Pattison AJ, McGarry MD, Perreard IM, Swienckowski JG, Eskey CJ, Lollis SS, Paulsen KD. Brain mechanical property measurement using MRE with intrinsic activation. Physics in medicine and biology 2012, 57(22):

6. MR measurements of intrinsic displacements
Phase Contrast, retrospectically cardiac gated
Strong velocity encoding gradients to allow tissue velocities smaller than 2.5cm/s to be recorded.
Retrospective gating to recover velocity images at 8 cardiac phases
Fourier integration can recover displacements
Maximum motion amplitude is in the first harmonic
Periodic velocity waveform
Vector velocity field
Weaver JB, Pattison AJ, McGarry MD, Perreard IM, Swienckowski JG, Eskey CJ, Lollis SS, Paulsen KD. Brain mechanical property measurement using MRE with intrinsic activation. Physics in medicine and biology 2012, 57(22):

7. 1Hz External Hydraulic Actuation System
measured displacement (mm)

8. 1Hz Actuation Phantom Results
Poroelastic
Results (Pa)
Viscoelastic
Results 1 (Pa)
T2 Weighted Image
15000
1500
5.5%
16mm
4.5%
9mm
7.5%
6.5%

9. IA-MRE: Meningioma Shear Modulus Elastogram (Pa) T2 Weighted MRI
6.5kPa
Poroelastic reconstruction
0Pa

10. IA-MRE: Metastatic Lung Cancer
T2 Weighted MRI
Shear Modulus Elastogram (Pa)
6.5kPa
Poroelastic reconstruction
0Pa

11. IA-MRE Anaplastic Oligodendroglioma
T2 Weighted MRI
Shear Modulus Elastogram (Pa)
20kPa
Poroelastic reconstruction
0Pa

12. Geisel School of Medicine at Dartmouth
Acknowledgements
Thayer School of Engineering at Dartmouth
Matthew D. McGarry, Ph.D
Scott Gordon. Ph.D.
Likun Tan, Ph.D
John B. Weaver Ph.D
Keith D. Paulsen, Ph.D
Geisel School of Medicine at Dartmouth
Heather Wishart, Ph.D.
Université de Sherbrooke
(Québec, Canada)
Elijah E. W. Van Houten, Ph.D
Dartmouth Hitchcock Medical Center
Lara K. Ronan, M.D
Jennifer Hong, M.D
Nathan Simmons, M.D
Angeline S. Andrew, Ph.D
Funding
NIBIB R01-EB
Hitchcock Grant

13. EXTRAS: Viscoelastic Mechanical model at low frequencies
𝛻⋅𝜇 𝛻𝑢+𝛻 𝑢 𝑇 +𝛻 𝜇𝜈 1−2𝜈 𝛻⋅𝑢=−𝜌 𝜔 2 𝑢
At low frequencies, 𝜌 𝜔 2 ≈0
𝛻⋅𝜇 𝛻𝑢+𝛻 𝑢 𝑇 +𝛻 𝜇𝜈 1−2𝜈 𝛻⋅𝑢=0
Viscoelastic
Single phase
Elastic energy storage
Viscous energy attenuation
Take out a factor of 𝜇 𝑜 , so that 𝜇 𝑥 = 𝜇 𝑜 𝜇 ∗ (𝑥)
𝜇 𝑜 𝛻⋅ 𝜇 ∗ 𝛻𝑢+𝛻 𝑢 𝑇 +𝛻 𝜇 ∗ 𝜈 1−2𝜈 𝛻⋅𝑢 =0
The solution, 𝑢(𝑥) is the same for any value of 𝜇 𝑜 , so there are multiple sets of 𝜇 ∗ (𝑥) that will give the exact same displacement solution. All 𝜇 ∗ (𝑥) solutions are a scaled version of the true 𝜇(𝑥).

14. EXTRAS:Poroelastic Mechanical model at low frequencies
Stress equation
𝛻⋅𝜇 𝛻𝑢+𝛻 𝑢 𝑇 +𝛻 𝜇𝜈 1−2𝜈 𝛻⋅𝑢− 1−𝛽 𝛻𝑃=−(𝜌−𝛽 𝜌 𝑓 ) 𝜔 2 𝑢+𝐹
𝛽= 𝜔 𝜙 2 𝜌 𝑓 𝜅 𝜔𝜅 𝜌 𝑎 +𝜙 𝜌 𝑓 +𝑖 𝜙 2
Pressure equation
𝛻⋅ 𝛽𝛻𝑃 + 𝜌 𝑓 𝜔 2 𝛻 1−𝛽 𝑢 =0
At low frequencies, 𝜔 2 ≈0
𝛻⋅𝜇 𝛻𝑢+𝛻 𝑢 𝑇 +𝛻 𝜇𝜈 1−2𝜈 𝛻⋅𝑢− 1−𝛽 𝛻𝑃=0
Biphasic: Fluid saturated porous matrix
Elastic properties of matrix
Hydrodynamic properties
𝛻⋅ 𝛽𝛻𝑃 =0
Take out a factor of 𝜇 𝑜 , so that 𝜇 𝑥 = 𝜇 𝑜 𝜇 ∗ (𝑥)
Poroelastic
𝜇 𝑜 𝛻⋅ 𝜇 ∗ 𝛻𝑢+𝛻 𝑢 𝑇 +𝛻 𝜇 ∗ 𝜈 1−2𝜈 𝛻⋅𝑢 − 1−𝛽 𝛻𝑃=0
The extra terms in poroelasticity avoid the non-uniqueness issue until very low frequencies where pore fluid pressure equilibrium is reached, 𝛻𝑃→0.

15. 1Hz Actuation Phantom Results
Poroelastic Results (Pa)
Viscoelastic Results (Pa)
T2 Weighted Image
12000
1500
13.9% Gelatin
3.8% Gelatin
3.8% Gelatin
(Soft)
13.9% Gelatin
(Stiff)
Viscoelastic Results
Poroelastic Results