Cortical thickness estimation of the proximal femur from multi-view Dual-Energy X-ray Absorptiometry (DXA)

Hip fracture is the leading cause of acute orthopaedic hospital admission amongst the elderly, with around a third of patients not surviving one year post-fracture. Although various preventative therapies are available, patient selection is difficult. The current state-of-the-art risk assessment tool (FRAX) ignores focal structural defects, such as cortical bone thinning, a critical component in characterizing hip fragility. Cortical thickness can be measured using CT, but this is expensive and involves a significant radiation dose. Instead, Dual-Energy X-ray Absorptiometry (DXA) is currently the preferred imaging modality for assessing hip fracture risk and is used routinely in clinical practice. Our ambition is to develop a tool to measure cortical thickness using multi-view DXA instead of CT. In this initial study, we work with digitally reconstructed radiographs (DRRs) derived from CT data as a surrogate for DXA scans: this enables us to compare directly the thickness estimates with the gold standard CT results. Our approach involves a model-based femoral shape reconstruction followed by a data-driven algorithm to extract numerous cortical thickness point estimates. In a series of experiments on the shaft and trochanteric regions of 48 proximal femurs, we validated our algorithm and established its performance limits using 20 views in the range 0°-171°: estimation errors were 0:19 ± 0:53mm (mean +/- one standard deviation). In a more clinically viable protocol using four views in the range 0°-51°, where no other bony structures obstruct the projection of the femur, measurement errors were -0:07 ± 0:79 mm. © 2013 SPIE.

The propagation of acoustic waves in a slowly varying duct with radially sheared axial mean flow

We consider the propagation of acoustic waves along a cylindrical duct carrying radially sheared axial mean flow, in which the duct radius is allowed to vary slowly along the axis. In previous work [A.J. Cooper & N. Peake, Journal of Fluid Mechanics 445 (2001) 207-234.] radially sheared axial mean flow with nonzero swirl in a slowly varying duct was considered, but in this paper we set the swirl to zero, thereby allowing simplification of the calculations of both the mean and unsteady flows. In this approach the acoustic wavenumber and corresponding eigenfunction are determined locally, while the wave amplitude is found by solving an evolution equation along the duct. Sample results are presented, including a case in which, perhaps surprisingly, the number of cut-on modes increases as the duct radius decreases. © 2013 Elsevier Ltd. All rights reserved.

Acoustic analogy in swirling mean flow applied to predict rotor trailing-edge noise

The present study aims at accounting for swirling mean flow effects on rotor trailing-edge noise. Indeed, the mean flow in between the rotor and the stator of the fan or of a compressor stage is highly swirling. The extension of Ffowcs-Williams & Hawkings' acoustic analogy in a medium at rest with moving surfaces and of Goldstein's acoustic analogy in a circular duct with uniform mean flow to a swirling mean flow in an annular duct is introduced. It is first applied to tonal noise. In most cases, the swirl modifies the pressure distribution downstream of the fan. In several configurations, when the swirl is rather close to a solid body swirl, it is often sufficient to apply a simple Doppler effect correction when predicting the duct modes in uniform mean flow in order to predict accurately the noise radiated with swirl. However, in other realistic configurations, the swirling mean-flow effect cannot be addressed using this simple Doppler effect correction. Second, a rotor trailing-edge noise model accounting for both the effects of the annular duct and the swirling mean flow is developed and applied to a realistic fan rotor with different swirling and sheared mean flows (and as a result different associated blade stagger angles). The benchmark cases are built from the Boeing 18-inch Fan Rig Broadband Noise Test. In all cases the swirling mean flow has an effect. In some cases the a simple Doppler effect may address it, but, in other realistic configurations our acoustic analogy with swirl is needed. © 2012 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc.

Swirling mean flow effect on fan-trailing edge broadband noise in a lined annular duct

The present study aims at investigating the effect of a swirling mean flow and a lined annular duct on rotor trailing-edge noise. The objectives are to investigate these effects on the eigenvalues and a tailored Green's function on one hand and on the realistic case of the fan trailing-edge noise on the other hand. Indeed, the mean flow in between the rotor and the stator of the fan is highly swirling. Moreover, interstage liners are used to reduce the noise produced by the fan stage. The extension of Ffowcs-Williams & Hawkings' acoustic analogy in a medium at rest with moving surfaces, of Goldstein's acoustic analogy in a hardwall circular duct with uniform mean flow and of Rienstra & Tester's Green's function in an annular lined duct with uniform mean flow to a swirling mean flow in an annular duct with liner is introduced. First, the eigenvalues and the Green's function are investigated showing a strong effect of the swirl and of the liner. Second, a rotor trailing-edge noise model accounting for both the effects of the annular duct with lined walls and the swirling mean flow is developed and applied to a realistic fan rotor with different swirling mean flows (and as a result different associated blade stagger angles). The benchmark cases are built from the Boeing 18-inch Fan Rig Broadband Noise Test. In all cases the swirling mean flow has a strong effect on the absolute noise level. The overall liner insertion loss is little changed by the swirl in the studied cases.

The acoustic analogy in an annular duct with swirling mean flow

This paper is concerned with modelling the effects of swirling flow on turbomachinery noise. We develop an acoustic analogy to predict sound generation in a swirling and sheared base flow in an annular duct, including the presence of moving solid surfaces to account for blade rows. In so doing we have extended a number of classical earlier results, including Ffowcs Williams & Hawkings' equation in a medium at rest with moving surfaces, and Lilley's equation for a sheared but non-swirling jet. By rearranging the Navier-Stokes equations we find a single equation, in the form of a sixth-order differential operator acting on the fluctuating pressure field on the left-hand side and a series of volume and surface source terms on the right-hand side; the form of these source terms depends strongly on the presence of swirl and radial shear. The integral form of this equation is then derived, using the Green's function tailored to the base flow in the (rigid) duct. As is often the case in duct acoustics, it is then convenient to move into temporal, axial and azimuthal Fourier space, where the Green's function is computed numerically. This formulation can then be applied to a number of turbomachinery noise sources. For definiteness here we consider the noise produced downstream when a steady distortion flow is incident on the fan from upstream, and compare our results with those obtained using a simplistic but commonly used Doppler correction method. We show that in all but the simplest case the full inclusion of swirl within an acoustic analogy, as described in this paper, is required. © 2013 Cambridge University Press.

Wave propagation in a slowly-varying duct with mean vortical swirling flow

The propagation of unsteady disturbances in a slowlyvarying cylindrical duct carrying mean swirling flow is investigated using a multiple-scales technique. This is applicable to turbomachinery flow behind a rotor stage when the swirl and axial velocities are of the same order. The presence of mean vorticity couples acoustic and vorticity equations which produces an eigenvalue problem that is not self-adjoint unlike that for irrotational mean flow. In order to determine the amplitude variation along the duct, an adjoint solution for the coupled system of equations is derived. The solution breaks down where a mode changes from cut on to cut off. In this region the amplitude is governed by a form of Airy's equation, and the effect of swirl is to introduce a small shift in the origin of the Airy function away from the turning-point location. The variation of axial wavenumber and amplitude along the duct is calculated. In hard-walled ducts mean swirl is shown to produce much larger amplitude variation along the duct compared with a nonswirling flow. Mean swirl also has a large effect in ducts with finite-impedance walls which differs depending on whether modes are co-rotating with the swirl or counter rotating. © 2001 by A.J. Cooper, Published by the American Institute of Aeronautics and Astronautics, Inc.