Multilayer Annulus Research Articles

Delamination of the Annulus Fibrosus of the Intervertebral Disc: Using a Bovine Tail Model to Examine Effect of Separation Rate.

The intervertebral disc (IVD) is a complex structure, and recent evidence suggests that separations or delamination between layers of the annulus may contribute to degeneration development, a common cause of low back pain The purpose of the present experiment was to quantify the mechanical response of the layer-adjoining interlamellar matrix at different rates of separation. Understanding the rate-dependency of the interlamellar matrix, or the adhesion between adjacent layers of the disc, is important as the spine experiences various loading velocities during activities of daily living. Twelve discs were dissected from four bovine tails (three extracts per tail). Two multi-layered annulus samples were collected from each IVD (total = 24, mean bond width = 3.82 ± 0.96 mm) and randomly assigned to a 180° peel test at one of three delamination rates; 0.05 mm/s, 0.5 mm/s, or 5 mm/s. Annulus extracts were found to have similar maximal adhesion strengths (p = 0.39) and stiffness (p = 0.97) across all rate conditions. However, a significant difference in lamellar adhesion strength variability was observed between the 5 mm/s condition (0.96 N/mm ± 0.31) when compared to the 0.5 mm/s (0.50 N/mm ± 0.19) and 0.05 mm/s (0.37 N/mm ± 0.13) conditions (p < 0.05). Increased variability may be indicative of non-uniform strength due to inconsistent adhesion throughout the interlamellar matrix, which is exacerbated by increased rates of loading. The observed non-uniform strength could possibly lead to a scenario more favourable to the development of microtrauma, and eventual delamination.

The Effect of Local Hydration Environment on the Mechanical Properties and Unloaded Temporal Changes of Isolated Porcine Annular Samples.

Preventing dehydration during in vitro testing of isolated layers of annulus fibrosus tissue may require different test conditions than functional spine units. The purpose of the study was twofold: (A) to quantify changes in mass and thickness of multilayer annulus samples in four hydration environments over 120 min; and (B) to quantify cycle-varying biaxial tensile properties of annulus samples in the four environments. The environments included a saline bath, air, relative humidity control, and misting combined with controlled humidity. The loading protocol implemented 24 cycles of biaxial tensile loading to 20% strain at a rate of 2%/s with 3-, 8-, and 13-min of intermittent rest. Specimen mass increased an average (standard deviation) 72% (11) when immersed for 120 min (p < 0.0001). The air condition and the combined mist and relative humidity conditions reduced mass by 45% (15) and 25% (23), respectively, after 120 min (p < 0.0014). Stress at 16% stretch in the air condition was higher at cycle 18 (18 min of exposure) and cycle 24 (33 min of exposure) compared to all other environments in both the axial and circumferential directions (p < 0.0460). There was no significant change in mass or thickness over time in the relative humidity condition and the change in circumferential stress at 16% stretch between cycles 6 and 24 was a maximum of 0.099 MPa and not statistically significant. Implementation of a controlled relative humidity environment is recommended to maintain hydration of isolated annulus layers during cyclic tensile testing.

Numerical Analysis of Thermal Development of Laminar Flow in a Concentric Multilayer Annulus

AbstractThe thermal behavior of laminar flow in a concentric multilayer annulus is investigated numerically for varying dimensionless inner radii and a defined number of fluid layers. Under a hydrodynamically developed flow assumption within the fluid layers, the development of temperature profiles and heat transfer along the annulus are analyzed for two different boundary conditions. The mean temperature distribution, local Nusselt number, and mean Nusselt number are discussed in detail with an emphasis on the effects of the inner radius and number of fluid layers. The obtained results indicate that the change in heat transfer coefficient in a layered annulus is more pronounced at a small inner radius or larger radius ratio. A further insertion of more than ten layers has insignificant influence on the convective heat transfer in a layered annulus.

Analytical solutions to time-fractional partial differential equations in a two-dimensional multilayer annulus

In this paper, analytical solutions to time-fractional partial differential equations in a multi-layer annulus are presented. The final solutions are obtained in terms of Mittag-Leffler function by using the finite integral transform technique and Laplace transform technique. In addition, the classical diffusion equation (α=1), the Helmholtz equation (α→0) and the wave equation (α=2) are discussed as special cases. Finally, an illustrative example problem for the three-layer semi-circular annular region is solved and numerical results are presented graphically for various kind of order of fractional derivative.

Finite integral transform method to solve asymmetric heat conduction in a multilayer annulus with time-dependent boundary conditions

The separation of variables ( SOV) method has recently been applied to solve time-dependent heat conduction problem in a multilayer annulus. It is observed that the transverse (radial) eigenvalues for the solution in polar ( r- θ) coordinate system are always real numbers (unlike in the case of similar multidimensional Cartesian problems where they may be imaginary) allowing one to obtain the solution analytically. However, the SOV method cannot be applied when the boundary conditions and/or the source terms are time-dependent, for example, in a nuclear fuel rod subjected to time-dependent boundaries or heat sources. In this paper, we present an alternative approach using the finite integral transform method to solve the asymmetric heat conduction problem in a multilayer annulus with time-dependent boundary conditions and/or heat sources. An eigenfunction expansion approach satisfying periodic boundary condition in the angular direction is used. After integral transformation and subsequent weighted summation over the radial layers, partial derivative with respect to r-variable is eliminated and, first order ordinary differential equations ( ODEs) are formed for the transformed temperatures. Solutions of ODEs are then inverted to obtain the temperature distribution in each layer. Since the proposed solution requires the same eigenfunctions as those in the similar problem with time-independent sources and/or boundary conditions—a problem solved using the SOV method—it is also “free” from imaginary eigenvalues.

A microstructural investigation of intervertebral disc lamellar connectivity: detailed analysis of the translamellar bridges

Little is known about the complex forces acting on the deformable multi-layered annulus at a microstructural level as the spine is compressed, flexed and twisted. The recently described translamellar bridging network radially linking many lamellae at discrete locations around the disc wall could be expected to play a significant biomechanical role. In this study, segments of annular wall that were sectioned at a range of angles (oblique, in-plane, sagittal and transverse) were examined using differential interference contrast microscopy to fully elucidate the fibrous detail of the translamellar bridging structures. Typically encompassing a width of 300-600 microm, translamellar bridging fibres proceed radially in the interbundle space within an individual lamella. Upon traversing the lamella, the bulk of these radial fibres bend through 90 degrees to merge with the fibres of the adjacent lamellae. The central fibres of this bridging system continue into the equivalent bridging structures in the adjacent lamellae. As well as exposing structural details that underpin the biomechanical properties of the disc wall, this study has also exposed the limitations of using standard section planes commonly employed by disc researchers.

Analytical Solution to Transient Asymmetric Heat Conduction in a Multilayer Annulus

In this paper, we present an analytical double-series solution for the time-dependent asymmetric heat conduction in a multilayer annulus. In general, analytical solutions in multidimensional Cartesian or cylindrical (r,z) coordinates suffer from existence of imaginary eigenvalues and thus may lead to numerical difficulties in computing analytical solution. In contrast, the proposed analytical solution in polar coordinates (2D cylindrical) is “free” from such imaginary eigenvalues. Real eigenvalues are obtained by virtue of precluded explicit dependence of transverse (radial) eigenvalues on those in the other direction.

A new design approach for direct embedment foundations (for power line supports)

An improved model has been developed for the analysis and design of directly embedded, single-pole transmission structures subject to high overturning moments. The model uses a multispring, nonlinear subgrade modulus approach to predict the load deflection response and ultimate capacity of direct embedment foundations placed in a multilayered soil subsurface profile, and with uniform or multilayered annulus backfill. To verify the predictive capabilities of the model, ten full-scale lateral load tests were conducted on directly embedded transmission poles. The model is described, and a comparison is made between the results of full-scale load tests and model predictions of the ultimate overturning moment capacity and load deflection behavior. The semiempirical model has been implemented in the computer program MFAD (Moment Foundation Analysis and Design) contained in EPRIs TLWorkstation. The results of 10 full-scale load tests demonstrate that the model, on the average, conservatively underpredicts ultimate moment capacity by approximately 20% and has a coefficient of variation of 12%. The model is also a good predictor of moment-deflection and moment-rotation response for well-compacted backfill materials, conservatively overpredicting deflection by approximately 16% and rotation by approximately 8%.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Heat Flow and Sonic Microstreaming

Measurements are being made to determine solid-to-liquid heat flow in the presence of sonically induced microstreaming near the interface. The microstreaming is induced by an air bubble vibrating near the region of heat flow. The air bubble is placed by means of a micropipette on the bottom of a clear plastic chamber containing the liquid. A 2-kc piston vibrator operated at low amplitudes a few centimeters above the bubble causes the bubble to become a strong source of sonic microstreaming. Imbedded in the plastic on which the bubble sits, and concentric with the bubble, is a thin multilayered annulus, the top of which is flush with the surface of the plastic. For a typical annulus the too layer is copper foil, the second is constantan; the bottom layer is a thermistor wafer through which current can be sent to produce a desired amount of heating. The thermal emf developed at the copper-constantan junction can be measured. Thus we have a source of known heat flow and known surface temperature. Investigations have been carried out in which heat flow was measured as a function of sonic amplitude, the surface temperature of the annulus being kept constant. The results indicate that, when the sonic amplitude A is small, the heat flow H varies linearly with A. This linear relationship between H and A is in agreement with the predictions of a simplified theory. At higher amplitudes the bubble may move about erratically, so that consistent readings are difficult to obtain. (Supported in part by the U. S. Air Force Office of Scientific Research.)