Frequency-dependent shear properties of annulus fibrosus and nucleus pulposus by magnetic resonance elastography.

Aging and degeneration are associated with changes in mechanical properties in the intervertebral disc, generating interest in the establishment of mechanical properties as early biomarkers for the degenerative cascade. Magnetic resonance elastography (MRE) of the intervertebral disc is usually limited to the nucleus pulposus, as the annulus fibrosus is stiffer and less hydrated. The objective of this work was to adapt high-frequency needle MRE to the characterization of the shear modulus of both the nucleus pulposus and annulus fibrosus. Bovine intervertebral discs were removed from fresh oxtails and characterized by needle MRE. The needle was inserted in the center of the disc and vibrations were generated by an amplified piezoelectric actuator. MRE acquisitions were performed on a 4.7-T small-animal MR scanner using a spin echo sequence with sinusoidal motion encoding gradients. Acquisitions were repeated over a frequency range of 1000-1800 Hz. The local frequency estimation inversion algorithm was used to compute the shear modulus. Stiffness maps allowed the visualization of the soft nucleus pulposus surrounded by the stiffer annulus fibrosus surrounded by the homogeneous gel. A significant difference in shear modulus between the nucleus pulposus and annulus fibrosus, and an increase in the shear modulus with excitation frequency, were observed, in agreement with the literature. This study demonstrates that global characterization of both the nucleus pulposus and annulus fibrosus of the intervertebral disc is possible with needle MRE using a preclinical magnetic resonance imaging (MRI) scanner. MRE can be a powerful method for the mapping of the complex properties of the intervertebral disc. The developed method could be adapted for in situ use by preserving adjacent vertebrae and puncturing the side of the intervertebral disc, thereby allowing an assessment of the contribution of osmotic pressure to the mechanical behavior of the intervertebral disc.

Requirements for accurate estimation of anisotropic material parameters by magnetic resonance elastography: A computational study.

PURPOSE: To establish the essential requirements for characterization of a transversely isotropic material by magnetic resonance elastography (MRE). THEORY AND METHODS: Three methods for characterizing nearly incompressible, transversely isotropic (ITI) materials were used to analyze data from closed-form expressions for traveling waves, finite-element (FE) simulations of waves in homogeneous ITI material, and FE simulations of waves in heterogeneous material. Key properties are the complex shear modulus µ2 , shear anisotropy ϕ=µ1/µ2-1, and tensile anisotropy ζ=E1/E2-1. RESULTS: Each method provided good estimates of ITI parameters when both slow and fast shear waves with multiple propagation directions were present. No method gave accurate estimates when the displacement field contained only slow shear waves, only fast shear waves, or waves with only a single propagation direction. Methods based on directional filtering are robust to noise and include explicit checks of propagation and polarization. Curl-based methods led to more accurate estimates in low noise conditions. Parameter estimation in heterogeneous materials is challenging for all methods. CONCLUSIONS: Multiple shear waves, both slow and fast, with different propagation directions, must be present in the displacement field for accurate parameter estimates in ITI materials. Experimental design and data analysis can ensure that these requirements are met. Magn Reson Med 78:2360-2372, 2017. © 2017 International Society for Magnetic Resonance in Medicine.

A cortical folding model incorporating stress-dependent growth explains gyral wavelengths and stress patterns in the developing brain.

In humans and many other mammals, the cortex (the outer layer of the brain) folds during development. The mechanics of folding are not well understood; leading explanations are either incomplete or at odds with physical measurements. We propose a mathematical model in which (i) folding is driven by tangential expansion of the cortex and (ii) deeper layers grow in response to the resulting stress. In this model the wavelength of cortical folds depends predictably on the rate of cortical growth relative to the rate of stress-induced growth. We show analytically and in simulations that faster cortical expansion leads to shorter gyral wavelengths; slower cortical expansion leads to long wavelengths or even smooth (lissencephalic) surfaces. No inner or outer (skull) constraint is needed to produce folding, but initial shape and mechanical heterogeneity influence the final shape. The proposed model predicts patterns of stress in the tissue that are consistent with experimental observations.

Elastic characterization of transversely isotropic soft materials by dynamic shear and asymmetric indentation.

The mechanical characterization of soft anisotropic materials is a fundamental challenge because of difficulties in applying mechanical loads to soft matter and the need to combine information from multiple tests. A method to characterize the linear elastic properties of transversely isotropic soft materials is proposed, based on the combination of dynamic shear testing (DST) and asymmetric indentation. The procedure was demonstrated by characterizing a nearly incompressible transversely isotropic soft material. A soft gel with controlled anisotropy was obtained by polymerizing a mixture of fibrinogen and thrombin solutions in a high field magnet (B = 11.7 T); fibrils in the resulting gel were predominantly aligned parallel to the magnetic field. Aligned fibrin gels were subject to dynamic (20-40 Hz) shear deformation in two orthogonal directions. The shear storage modulus was 1.08 ± 0. 42 kPa (mean ± std. dev.) for shear in a plane parallel to the dominant fiber direction, and 0.58 ± 0.21 kPa for shear in the plane of isotropy. Gels were indented by a rectangular tip of a large aspect ratio, aligned either parallel or perpendicular to the normal to the plane of transverse isotropy. Aligned fibrin gels appeared stiffer when indented with the long axis of a rectangular tip perpendicular to the dominant fiber direction. Three-dimensional numerical simulations of asymmetric indentation were used to determine the relationship between direction-dependent differences in indentation stiffness and material parameters. This approach enables the estimation of a complete set of parameters for an incompressible, transversely isotropic, linear elastic material.

Propulsive forces on the flagellum during locomotion of Chlamydomonas reinhardtii.

The distributed propulsive forces exerted on the flagellum of the swimming alga Chlamydomonas reinhardtii by surrounding fluid were estimated from experimental image data. Images of uniflagellate mutant Chlamydomonas cells were obtained at 350 frames/s with 125-nm spatial resolution, and the motion of the cell body and the flagellum were analyzed in the context of low-Reynolds-number fluid mechanics. Wild-type uniflagellate cells, as well as uniflagellate cells lacking inner dynein arms (ida3) or outer dynein arms (oda2) were studied. Ida3 cells exhibit stunted flagellar waveforms, whereas oda2 cells beat with lower frequency. Image registration and sorting algorithms provided high-resolution estimates of the motion of the cell body, as well as detailed kinematics of the flagellum. The swimming cell was modeled as an ellipsoid in Stokes flow, propelled by viscous forces on the flagellum. The normal and tangential components of force on the flagellum (f(N) and f(T)) were related by resistive coefficients (C(N) and C(T)) to the corresponding components of velocity (V(N) and V(T)).The values of these coefficients were estimated by satisfying equilibrium requirements for force and torque on the cell. The estimated values of the resistive coefficients are consistent among all three genotypes and similar to theoretical predictions.

Natural oscillatory modes of 3D deformation of the human brain in vivo.

Natural modes and frequencies of three-dimensional (3D) deformation of the human brain were identified from in vivo tagged magnetic resonance images (MRI) acquired dynamically during transient mild acceleration of the head. Twenty 3D strain fields, estimated from tagged MRI image volumes in 19 adult subjects, were analyzed using dynamic mode decomposition (DMD). These strain fields represented dynamic, 3D brain deformations during constrained head accelerations, either involving rotation about the vertical axis of the neck or neck extension. DMD results reveal fundamental oscillatory modes of deformation at damped frequencies near 7 Hz (in neck rotation) and 11 Hz (in neck extension). Modes at these frequencies were found consistently among all subjects. These characteristic features of 3D human brain deformation are important for understanding the response of the brain in head impacts and provide valuable quantitative criteria for the evaluation and use of computer models of brain mechanics.

Measurement of anisotropic mechanical properties in porcine brain white matter ex vivo using magnetic resonance elastography.

The mechanical properties of brain tissue, particularly those of white matter (WM), need to be characterized accurately for use in finite element (FE) models of brain biomechanics and traumatic brain injury (TBI). Magnetic resonance elastography (MRE) is a powerful tool for non-invasive estimation of the mechanical properties of soft tissues. While several studies involving direct mechanical tests of brain tissue have shown mechanical anisotropy, most MRE studies of brain tissue assume an isotropic model. In this study, an incompressible transversely isotropic (TI) material model parameterized by minimum shear modulus (µ2), shear anisotropy parameter (ϕ), and tensile anisotropy parameter (ζ) is applied to analyze MRE measurements of ex vivo porcine white matter (WM) brain tissue. To characterize shear anisotropy, "slow" (pure transverse) shear waves were propagated at 100, 200 and 300Hz through sections of ex vivo brain tissue including both WM and gray matter (GM). Shear waves were found to propagate with elliptical fronts, consistent with TI material behavior. Shear wave fields were also analyzed within regions of interest (ROI) to find local shear wavelengths parallel and perpendicular to fiber orientation. FE simulations of a TI material with a range of plausible shear modulus (µ2) and shear anisotropy parameters (ϕ) were run and the results were analyzed in the same fashion as the experimental case. Parameters of the FE simulations which most closely matched each experiment were taken to represent the mechanical properties of that particular sample. Using this approach, WM in the ex vivo porcine brain was found to be mildly anisotropic in shear with estimates of minimum shear modulus (actuation frequencies listed in parenthesis): µ2= 1.04 ± 0.12 kPa (at 100Hz), µ2= 1.94 ± 0.29 kPa (at 200Hz), and µ2= 2.88 ± 0.34 kPa (at 300Hz) and corresponding shear anisotropy factors of ϕ= 0.27 ± 0.09 (at 100Hz), ϕ= 0.29 ± 0.14 (at 200Hz) and ϕ= 0.34 ± 0.13 (at 300Hz). Future MRE studies will focus on tensile anisotropy, which will require both slow and fast shear waves for accurate estimation.

A longitudinal magnetic resonance elastography study of murine brain tumors following radiation therapy.

An accurate and noninvasive method for assessing treatment response following radiotherapy is needed for both treatment monitoring and planning. Measurement of solid tumor volume alone is not sufficient for reliable early detection of therapeutic response, since changes in physiological and/or biomechanical properties can precede tumor volume change following therapy. In this study, we use magnetic resonance elastography to evaluate the treatment effect after radiotherapy in a murine brain tumor model. Shear modulus was calculated and compared between the delineated tumor region of interest (ROI) and its contralateral, mirrored counterpart. We also compared the shear modulus from both the irradiated and non-irradiated tumor and mirror ROIs longitudinally, sampling four time points spanning 9-19 d post tumor implant. Results showed that the tumor ROI had a lower shear modulus than that of the mirror ROI, independent of radiation. The shear modulus of the tumor ROI decreased over time for both the treated and untreated groups. By contrast, the shear modulus of the mirror ROI appeared to be relatively constant for the treated group, while an increasing trend was observed for the untreated group. The results provide insights into the tumor properties after radiation treatment and demonstrate the potential of using the mechanical properties of the tumor as a biomarker. In future studies, more closely spaced time points will be employed for detailed analysis of the radiation effect.

Magnetic resonance elastography of slow and fast shear waves illuminates differences in shear and tensile moduli in anisotropic tissue.

Mechanical anisotropy is an important property of fibrous tissues; for example, the anisotropic mechanical properties of brain white matter may play a key role in the mechanics of traumatic brain injury (TBI). The simplest anisotropic material model for small deformations of soft tissue is a nearly incompressible, transversely isotropic (ITI) material characterized by three parameters: minimum shear modulus (µ), shear anisotropy (ϕ=µ1µ-1) and tensile anisotropy (ζ=E1E2-1). These parameters can be determined using magnetic resonance elastography (MRE) to visualize shear waves, if the angle between the shear-wave propagation direction and fiber direction is known. Most MRE studies assume isotropic material models with a single shear (µ) or tensile (E) modulus. In this study, two types of shear waves, "fast" and "slow", were analyzed for a given propagation direction to estimate anisotropic parameters µ, ϕ, and ζ in two fibrous soft materials: turkey breast ex vivo and aligned fibrin gels. As expected, the speed of slow shear waves depended on the angle between fiber direction and propagation direction. Fast shear waves were observed when the deformations due to wave motion induced stretch in the fiber direction. Finally, MRE estimates of anisotropic mechanical properties in turkey breast were compared to estimates from direct mechanical tests.

Viscoelastic properties of the ferret brain measured in vivo at multiple frequencies by magnetic resonance elastography.

Characterization of the dynamic mechanical behavior of brain tissue is essential for understanding and simulating the mechanisms of traumatic brain injury (TBI). Changes in mechanical properties may also reflect changes in the brain due to aging or disease. In this study, we used magnetic resonance elastography (MRE) to measure the viscoelastic properties of ferret brain tissue in vivo. Three-dimensional (3D) displacement fields were acquired during wave propagation in the brain induced by harmonic excitation of the skull at 400 Hz, 600 Hz and 800 Hz. Shear waves with wavelengths in the order of millimeters were clearly visible in the displacement field, in strain fields, and in the curl of displacement field (which contains no contributions from longitudinal waves). Viscoelastic parameters (storage and loss moduli) governing dynamic shear deformation were estimated in gray and white matter for these excitation frequencies. To characterize the reproducibility of measurements, two ferrets were studied on three different dates each. Estimated viscoelastic properties of white matter in the ferret brain were generally similar to those of gray matter and consistent between animals and scan dates. In both tissue types G' increased from approximately 3 kPa at 400 Hz to 7 kPa at 800 Hz and G″ increased from approximately 1 kPa at 400 Hz to 2 kPa at 800 Hz. These measurements of shear wave propagation in the ferret brain can be used to both parameterize and validate finite element models of brain biomechanics.

Viscoelastic properties of soft gels: comparison of magnetic resonance elastography and dynamic shear testing in the shear wave regime.

Magnetic resonance elastography (MRE) is used to quantify the viscoelastic shear modulus, G*, of human and animal tissues. Previously, values of G* determined by MRE have been compared to values from mechanical tests performed at lower frequencies. In this study, a novel dynamic shear test (DST) was used to measure G* of a tissue-mimicking material at higher frequencies for direct comparison to MRE. A closed-form solution, including inertial effects, was used to extract G* values from DST data obtained between 20 and 200 Hz. MRE was performed using cylindrical 'phantoms' of the same material in an overlapping frequency range of 100-400 Hz. Axial vibrations of a central rod caused radially propagating shear waves in the phantom. Displacement fields were fit to a viscoelastic form of Navier's equation using a total least-squares approach to obtain local estimates of G*. DST estimates of the storage G' (Re[G*]) and loss modulus G″ (Im[G*]) for the tissue-mimicking material increased with frequency from 0.86 to 0.97 kPa (20-200 Hz, n = 16), while MRE estimates of G' increased from 1.06 to 1.15 kPa (100-400 Hz, n = 6). The loss factor (Im[G*]/Re[G*]) also increased with frequency for both test methods: 0.06-0.14 (20-200 Hz, DST) and 0.11-0.23 (100-400 Hz, MRE). Close agreement between MRE and DST results at overlapping frequencies indicates that G* can be locally estimated with MRE over a wide frequency range. Low signal-to-noise ratio, long shear wavelengths and boundary effects were found to increase residual fitting error, reinforcing the use of an error metric to assess confidence in local parameter estimates obtained by MRE.

Relative brain displacement and deformation during constrained mild frontal head impact.

This study describes the measurement of fields of relative displacement between the brain and the skull in vivo by tagged magnetic resonance imaging and digital image analysis. Motion of the brain relative to the skull occurs during normal activity, but if the head undergoes high accelerations, the resulting large and rapid deformation of neuronal and axonal tissue can lead to long-term disability or death. Mathematical modelling and computer simulation of acceleration-induced traumatic brain injury promise to illuminate the mechanisms of axonal and neuronal pathology, but numerical studies require knowledge of boundary conditions at the brain-skull interface, material properties and experimental data for validation. The current study provides a dense set of displacement measurements in the human brain during mild frontal skull impact constrained to the sagittal plane. Although head motion is dominated by translation, these data show that the brain rotates relative to the skull. For these mild events, characterized by linear decelerations near 1.5g (g = 9.81 m s⁻²) and angular accelerations of 120-140 rad s⁻², relative brain-skull displacements of 2-3 mm are typical; regions of smaller displacements reflect the tethering effects of brain-skull connections. Strain fields exhibit significant areas with maximal principal strains of 5 per cent or greater. These displacement and strain fields illuminate the skull-brain boundary conditions, and can be used to validate simulations of brain biomechanics.

Effect of residual stress and heterogeneity on circumferential stress in the arterial wall.

Quantifying the stress distribution through the arterial wall is essential to studies of arterial growth and disease. Previous studies have shown that both residual stress, as measured by opening angle, and differing material properties for the media-intima and the adventitial layers affect the transmural circumferential stress (sigma theta) distribution. Because a lack of comprehensive data on a single species and artery has led to combinations from multiple sources, this study determined the sensitivity of sigma theta to published variations in both opening angle and layer thickness data. We fit material properties to previously published experimental data for pressure-diameter relations and opening angles of rabbit carotid artery, and predicted sigma theta through the arterial wall at physiologic conditions. Using a one-layer model, the ratio of sigma theta at the internal wall to the mean sigma theta decreased from 2.34 to 0.98 as the opening angle increased from 60 to 130 deg. In a two-layer model using a 95 deg opening angle, mean sigma theta in the adventitia increased (112 percent for 25 percent adventitia) and mean sigma theta in the media decreased (47 percent for 25 percent adventitia). These results suggest that both residual stress and wall layers have important effects on transmural stress distribution. Thus, experimental measurements of loading curves, opening angles, and wall composition from the same species and artery are needed to accurately predict the transmural stress distribution in the arterial wall.

Measurement of strain in physical models of brain injury: a method based on HARP analysis of tagged magnetic resonance images (MRI).

Two-dimensional (2-D) strain fields were estimated non-invasively in two simple experimental models of closed-head brain injury. In the first experimental model, shear deformation of a gel was induced by angular acceleration of its spherical container In the second model the brain of a euthanized rat pup was deformed by indentation of its skull. Tagged magnetic resonance images (MRI) were obtained by gated image acquisition during repeated motion. Harmonic phase (HARP) images corresponding to the spectral peaks of the original tagged MRI were obtained, following procedures proposed by Osman, McVeigh and Prince. Two methods of HARP strain analysis were applied, one based on the displacement of tag line intersections, and the other based on the gradient of harmonic phase. Strain analysis procedures were also validated on simulated images of deformed grids. Results show that it is possible to visualize deformation and to quantify strain efficiently in animal models of closed head injury.

Epicardial suction: a new approach to mechanical testing of the passive ventricular wall.

The lack of an appropriate three-dimensional constitutive relation for stress in passive ventricular myocardium currently limits the utility of existing mathematical models for experimental and clinical applications. Previous experiments used to estimate parameters in three-dimensional constitutive relations, such as biaxial testing of excised myocardial sheets or passive inflation of the isolated arrested heart, have not included significant transverse shear deformation or in-plane compression. Therefore, a new approach has been developed in which suction is applied locally to the ventricular epicardium to introduce a complex deformation in the region of interest, with transmural variations in the magnitude and sign of nearly all six strain components. The resulting deformation is measured throughout the region of interest using magnetic resonance tagging. A nonlinear, three-dimensional, finite element model is used to predict these measurements at several suction pressures. Parameters defining the material properties of this model are optimized by comparing the measured and predicted myocardial deformations. We used this technique to estimate material parameters of the intact passive canine left ventricular free wall using an exponential, transversely isotropic constitutive relation. We tested two possible models of the heart wall: first, that it was homogeneous myocardium, and second, that the myocardium was covered with a thin epicardium with different material properties. For both models, in agreement with previous studies, we found that myocardium was nonlinear and anisotropic with greater stiffness in the fiber direction. We obtained closer agreement to previously published strain data from passive filling when the ventricular wall was modeled as having a separate, isotropic epicardium. These results suggest that epicardium may play a significant role in passive ventricular mechanics.