Spatially-resolved hydraulic conductivity estimation via poroelastic magnetic resonance elastography

On the relationship between viscoelasticity and water diffusion in soft biological tissues

Magnetic resonance elastography (MRE) and diffusion-weighted imaging (DWI) are complementary imaging techniques that detect disease based on viscoelasticity and water mobility, respectively. However, the relationship between viscoelasticity and water diffusion is still poorly understood, hindering the clinical translation of combined DWI-MRE markers. We used DWI-MRE to study 129 biomaterial samples including native and cross-linked collagen, glycosaminoglycans (GAGs) with different sulfation levels, and decellularized specimens of pancreas and liver, all with different proportions of solid tissue, or solid fractions. We developed a theoretical framework of the relationship between mechanical loss and tissue-water mobility based on two parameters, solid and fluid viscosity. These parameters revealed distinct DWI-MRE property clusters characterizing weak, moderate, and strong water-network interactions. Sparse networks interacting weakly with water, such as collagen or diluted decellularized tissue, resulted in marginal changes in water diffusion over increasing solid viscosity. In contrast, dense networks with larger solid fractions exhibited both free and hindered water diffusion depending on the polarity of the solid components. For example, polar and highly sulfated GAGs as well as native soft tissues hindered water diffusion despite relatively low solid viscosity. Our results suggest that two fundamental properties of tissue networks, solid fraction and network polarity, critically influence solid and fluid viscosity in biological tissues. Since clinical DWI and MRE are sensitive to these viscosity parameters, the framework we present here can be used to detect tissue remodeling and architectural changes in the setting of diagnostic imaging. STATEMENT OF SIGNIFICANCE: The viscoelastic properties of biological tissues provide a wealth of information on the vital state of cells and host matrix. Combined measurement of viscoelasticity and water diffusion by medical imaging is sensitive to tissue microarchitecture. However, the relationship between viscoelasticity and water diffusion is still poorly understood, hindering full exploitation of these properties as a combined clinical biomarker. Therefore, we analyzed the parameter space accessible by diffusion-weighted imaging (DWI) and magnetic resonance elastography (MRE) and developed a theoretical framework for the relationship between water mobility and mechanical parameters in biomaterials. Our theory of solid material properties related to particle motion can be translated to clinical radiology using clinically established MRE and DWI.

Poroelastic Mechanical Properties of the Brain Tissue of Normal Pressure Hydrocephalus Patients During Lumbar Drain Treatment Using Intrinsic Actuation MR Elastography

RATIONALE AND OBJECTIVES

Hydrocephalus (HC) is caused by accumulating cerebrospinal fluid resulting in enlarged ventricles and neurological symptoms. HC can be treated via a shunt in a subset of patients; identifying which individuals will respond through noninvasive imaging would avoid complications from unsuccessful treatments. This preliminary work is a longitudinal study applying MR Elastography (MRE) to HC patients with a focus on normal pressure hydrocephalus (NPH).

MATERIALS AND METHODS

Twenty-two ventriculomegaly patients were imaged and subsequently received a lumbar drain placement for cerebrospinal fluid (CSF) drainage. NPH lumbar drain responders and NPH syndrome nonresponders were categorized by clinical presentation. Displacement images were acquired using intrinsic activation (IA) MRE and poroelastic inversion recovered shear stiffness and hydraulic conductivity values. A stable IA-MRE inversion protocol was developed to produce unique solutions for both recovered properties, independent of initial estimates.

RESULTS

Property images showed significantly increased shear modulus (p = 0.003 in periventricular region, p = 0.005 in remaining cerebral tissue) and hydraulic conductivity (p = 0.04 in periventricular region) in ventriculomegaly patients compared to healthy volunteers. Baseline MRE imaging did not detect significant differences between NPH lumbar drain responders and NPH syndrome nonresponders; however, MRE time series analysis demonstrated consistent trends in average poroelastic shear modulus values over the course of the lumbar drain process in responders (initial increase, followed by a later decrease) which did not occur in nonresponders.

CONCLUSION

These findings are indicative of acute mechanical changes in the brain resulting from CSF drainage in NPH patients.

Poroelasticity as a Model of Soft Tissue Structure: Hydraulic Permeability Reconstruction for Magnetic Resonance Elastography in Silico

Magnetic Resonance Elastography allows noninvasive visualization of tissue mechanical properties by measuring the displacements resulting from applied stresses, and fitting a mechanical model. Poroelasticity naturally lends itself to describing tissue - a biphasic medium, consisting of both solid and fluid components. This article reviews the theory of poroelasticity, and shows that the spatial distribution of hydraulic permeability, the ease with which the solid matrix permits the flow of fluid under a pressure gradient, can be faithfully reconstructed without spatial priors in simulated environments. The paper describes an in-house MRE computational platform - a multi-mesh, finite element poroelastic solver coupled to an artificial epistemic agent capable of running Bayesian inference to reconstruct inhomogenous model mechanical property images from measured displacement fields. Building on prior work, the domain of convergence for inference is explored, showing that hydraulic permeabilities over several orders of magnitude can be reconstructed given very little prior knowledge of the true spatial distribution.

Nonlinear Inversion MR Elastography With Low-Frequency Actuation

Magnetic resonance elastography (MRE) has been developed to noninvasively reconstruct mechanical properties for tissue and tissue-like materials over a frequency range of 10 ~200 Hz. In this work, low frequency (1~1.5 Hz) MRE activations were employed to estimate mechanical property distributions of simulated data and experimental phantoms. Nonlinear inversion (NLI) MRE algorithms based on viscoelastic and poroelastic material models were used to solve the inverse problems and recover images of the shear modulus and hydraulic conductivity. Data from a simulated phantom containing an inclusion with property contrast was carried out to study the feasibility of our low frequency actuated approach. To verify the stability of NLI algorithms for low frequency actuation, different levels of synthetic noise were added to the displacement data. Spatial distributions and property values were recovered well for noise level less than 5%. For the presented experimental phantom reconstructions with regularizations, the computed storage moduli from viscoelastic and poroelastic MRE gave similar results. Contrast was detected between inclusions and background in recovered hydraulic conductivity images. Results and findings confirm the feasibility of future in vivo neuroimaging examinations using natural cerebrovascular pulsations at cardiac frequencies, which can eliminate specialized equipment for high frequency actuation.

A poro-hyper-viscoelastic rate-dependent constitutive modeling for the analysis of brain tissues

In this paper, the dynamic behavior of bovine brain tissue, measured from in-vitro unconfined compression tests, is examined and represented through a viscoelastic biphasic model. The experiments have been carried out under three compression speeds of 10, 100, and 1000 mm/s. The results exhibited significant rate-dependent behavior. The brain tissue is modeled as a biphasic continuum consisting of a compressible solid matrix, fully saturated with an incompressible interstitial fluid. The governing equations based on conservation of mass and momentum are used to describe the solid-fluid interactions. An inverse scheme is employed in which a finite element model runs iteratively to optimize constitutive constants. The obtained material parameters of the proposed biphasic model show relatively good agreement (R² ≥ 0.96) with the experimental tissue mechanical responses at different rates. The model can successfully capture the key aspects of the rate-dependency for both solid and fluid phases under large strain deformation. This poro-hyper viscoelastic model can effectively estimate the global and local rate-dependent tissue deformations, the spatial variations in pore spaces, hydrostatic pressure as well as fluid diffusion through the tissue.

Reliable preparation of agarose phantoms for use in quantitative magnetic resonance elastography

Agarose phantoms are one type of phantom commonly used in developing in vivo brain magnetic resonance elastography (MRE) sequences because they are inexpensive and easy to work with, store, and dispose of; however, protocols for creating agarose phantoms are non-standardized and often result in inconsistent phantoms with significant variability in mechanical properties. Many magnetic resonance imaging (MRI) and ultrasound studies use phantoms, but often these phantoms are not tailored for desired mechanical properties and as such are too stiff or not mechanically consistent enough to be used in MRE. In this work, we conducted a systematic study of agarose phantom creation parameters to identify those factors that are most conducive to producing mechanically consistent agarose phantoms for MRE research. We found that cooling rate and liquid temperature affected phantom homogeneity. Phantom stiffness is affected by agar concentration (quadratically), by final liquid temperature and salt content in phantoms, and by the interaction of these two metrics each with stir rate. We captured and quantified the implied relationships with a regression model that can be used to estimate stiffness of resulting phantoms. Additionally, we characterized repeatability, stability over time, impact on MR signal parameters, and differences in agar gel microstructure. This protocol and regression model should prove beneficial in future MRE development studies that use phantoms to determine stiffness measurement accuracy.

MR elastography at 1 Hz of gelatin phantoms using 3D or 4D acquisition

Magnetic Resonance Elastography (MRE) detects induced periodic motions in biological tissues allowing maps of tissue mechanical properties to be derived. In-vivo MRE is commonly performed at frequencies of 30-100 Hz using external actuation, however, using cerebro-vascular pulsation at 1 Hz as a form of intrinsic actuation (IA-MRE) eliminates the need for external motion sources and simplifies data acquisition. In this study a hydraulic actuation system was developed to drive 1 Hz motions in gelatin as a tool for investigating the performance limits of IA-MRE image reconstruction under controlled conditions. Quantitative flow (QFLOW) MR techniques were used to phase encode 1 Hz motions as a function of gradient direction using 3D or 4D acquisition; 4D acquisition was twice as fast and yielded comparable motion field and concomitant image reconstruction results provided the motion signal was sufficiently strong. Per voxel motion noise floor corresponded to a displacement amplitude of about 20-30 μm. Signal to noise ratio (SNR) was 94 ± 17 for 3D and dropped to 69 ± 10 for the faster 4D acquisition, but yielded octahedral shear stress and shear modulus maps of high quality that differed by only about 20% on average. QFLOW measurements in gel phantoms were improved significantly by adding Mn(II) to mimic relaxation rates found in brain. Overall, the hydraulic 1 Hz actuation system when coupled with 4D sequence acquisition produced a fast reliable approach for future IA-MRE phantom evaluation and contrast detail studies needed to benchmark imaging performance.

A numerical framework for interstitial fluid pressure imaging in poroelastic MRE

A numerical framework for interstitial fluid pressure imaging (IFPI) in biphasic materials is investigated based on three-dimensional nonlinear finite element poroelastic inversion. The objective is to reconstruct the time-harmonic pore-pressure field from tissue excitation in addition to the elastic parameters commonly associated with magnetic resonance elastography (MRE). The unknown pressure boundary conditions (PBCs) are estimated using the available full-volume displacement data from MRE. A subzone-based nonlinear inversion (NLI) technique is then used to update mechanical and hydrodynamical properties, given the appropriate subzone PBCs, by solving a pressure forward problem (PFP). The algorithm was evaluated on a single-inclusion phantom in which the elastic property and hydraulic conductivity images were recovered. Pressure field and material property estimates had spatial distributions reflecting their true counterparts in the phantom geometry with RMS errors around 20% for cases with 5% noise, but degraded significantly in both spatial distribution and property values for noise levels > 10%. When both shear moduli and hydraulic conductivity were estimated along with the pressure field, property value error rates were as high as 58%, 85% and 32% for the three quantities, respectively, and their spatial distributions were more distorted. Opportunities for improving the algorithm are discussed.

Gradient-Based Optimization for Poroelastic and Viscoelastic MR Elastography

We describe an efficient gradient computation for solving inverse problems arising in magnetic resonance elastography (MRE). The algorithm can be considered as a generalized 'adjoint method' based on a Lagrangian formulation. One requirement for the classic adjoint method is assurance of the self-adjoint property of the stiffness matrix in the elasticity problem. In this paper, we show this property is no longer a necessary condition in our algorithm, but the computational performance can be as efficient as the classic method, which involves only two forward solutions and is independent of the number of parameters to be estimated. The algorithm is developed and implemented in material property reconstructions using poroelastic and viscoelastic modeling. Various gradient- and Hessian-based optimization techniques have been tested on simulation, phantom and in vivo brain data. The numerical results show the feasibility and the efficiency of the proposed scheme for gradient calculation.

Viscoelasticity of subcortical gray matter structures

Viscoelastic mechanical properties of the brain assessed with magnetic resonance elastography (MRE) are sensitive measures of microstructural tissue health in neurodegenerative conditions. Recent efforts have targeted measurements localized to specific neuroanatomical regions differentially affected in disease. In this work, we present a method for measuring the viscoelasticity in subcortical gray matter (SGM) structures, including the amygdala, hippocampus, caudate, putamen, pallidum, and thalamus. The method is based on incorporating high spatial resolution MRE imaging (1.6 mm isotropic voxels) with a mechanical inversion scheme designed to improve local measures in pre-defined regions (soft prior regularization [SPR]). We find that in 21 healthy, young volunteers SGM structures differ from each other in viscoelasticity, quantified as the shear stiffness and damping ratio, but also differ from the global viscoelasticity of the cerebrum. Through repeated examinations on a single volunteer, we estimate the uncertainty to be between 3 and 7% for each SGM measure. Furthermore, we demonstrate that the use of specific methodological considerations-higher spatial resolution and SPR-both decrease uncertainty and increase sensitivity of the SGM measures. The proposed method allows for reliable MRE measures of SGM viscoelasticity for future studies of neurodegenerative conditions. Hum Brain Mapp 37:4221-4233, 2016. © 2016 Wiley Periodicals, Inc.

Hydrocephalus in children

Hydrocephalus is a common disorder of cerebral spinal fluid (CSF) physiology resulting in abnormal expansion of the cerebral ventricles. Infants commonly present with progressive macrocephaly whereas children older than 2 years generally present with signs and symptoms of intracranial hypertension. The classic understanding of hydrocephalus as the result of obstruction to bulk flow of CSF is evolving to models that incorporate dysfunctional cerebral pulsations, brain compliance, and newly characterised water-transport mechanisms. Hydrocephalus has many causes. Congenital hydrocephalus, most commonly involving aqueduct stenosis, has been linked to genes that regulate brain growth and development. Hydrocephalus can also be acquired, mostly from pathological processes that affect ventricular outflow, subarachnoid space function, or cerebral venous compliance. Treatment options include shunt and endoscopic approaches, which should be individualised to the child. The long-term outcome for children that have received treatment for hydrocephalus varies. Advances in brain imaging, technology, and understanding of the pathophysiology should ultimately lead to improved treatment of the disorder.

Suitability of poroelastic and viscoelastic mechanical models for high and low frequency MR elastography

PURPOSE

Descriptions of the structure of brain tissue as a porous cellular matrix support application of a poroelastic (PE) mechanical model which includes both solid and fluid phases. However, the majority of brain magnetic resonance elastography (MRE) studies use a single phase viscoelastic (VE) model to describe brain tissue behavior, in part due to availability of relatively simple direct inversion strategies for mechanical property estimation. A notable exception is low frequency intrinsic actuation MRE, where PE mechanical properties are imaged with a nonlinear inversion algorithm.

METHODS

This paper investigates the effect of model choice at each end of the spectrum of in vivo human brain actuation frequencies. Repeat MRE examinations of the brains of healthy volunteers were used to compare image quality and repeatability for each inversion model for both 50 Hz externally produced motion and ≈1 Hz intrinsic motions. Additionally, realistic simulated MRE data were generated with both VE and PE finite element solvers to investigate the effect of inappropriate model choice for ideal VE and PE materials.

RESULTS

In vivo, MRE data revealed that VE inversions appear more representative of anatomical structure and quantitatively repeatable for 50 Hz induced motions, whereas PE inversion produces better results at 1 Hz. Reasonable VE approximations of PE materials can be derived by equating the equivalent wave velocities for the two models, provided that the timescale of fluid equilibration is not similar to the period of actuation. An approximation of the equilibration time for human brain reveals that this condition is violated at 1 Hz but not at 50 Hz. Additionally, simulation experiments when using the "wrong" model for the inversion demonstrated reasonable shear modulus reconstructions at 50 Hz, whereas cross-model inversions at 1 Hz were poor quality. Attenuation parameters were sensitive to changes in the forward model at both frequencies, however, no spatial information was recovered because the mechanisms of VE and PE attenuation are different.

CONCLUSIONS

VE inversions are simpler with fewer unknown properties and may be sufficient to capture the mechanical behavior of PE brain tissue at higher actuation frequencies. However, accurate modeling of the fluid phase is required to produce useful mechanical property images at the lower frequencies of intrinsic brain motions.