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

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.

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.

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

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): 7275-7287.

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): 7275-7287.

1Hz External Hydraulic Actuation System measured displacement (mm)

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%

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

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

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

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-EB018230-01 Hitchcock Grant

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).

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.

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