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Vector tomographic X-ray phase contrast velocimetry utilizing dynamic blood speckle

Sarah C. Irvine, David M. Paganin, R. Aidan Jamison, Stephen Dubsky, and Andreas Fouras

Open Access Open Access

Abstract

We present a time-resolved tomographic reconstruction of the velocity field associated with pulsatile blood flow through a rotationally-symmetric stenotic vessel model. The in-vitro sample was imaged using propagation-based phase contrast with monochromated X-rays from a synchrotron undulator source, and a fast shutter-synchronized detector with high-resolution used to acquire frames of the resulting dynamic speckle pattern. Having used phase retrieval to decode the phase contrast from the speckle patterns, the resulting projected-density maps were analysed using the statistical correlation methods of particle image velocimetry (PIV). This yields the probability density functions of blood-cell displacement within the vessel. The axial velocity-field component of the rotationally-symmetric flow was reconstructed using an inverse-Abel transform. A modified inverse-Abel transform was used to reconstruct the radial component. This vector tomographic phase-retrieval velocimetry was performed over the full pumping cycle, to completely characterize the velocity field of the pulsatile blood flow in both space and time.

©2010 Optical Society of America

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Figures (9)
Fig. 1
Fig. 1 Example of dynamic X-ray phase-contrast blood speckle, obtained when a blood-filled vessel approximately 2mm thick (inner diameter) is illuminated by 25 keV X-rays, with the resulting exit wavefield propagating 45cm to the detector plane. Field of view is 128×128 pixels = 115×115 μm (0.9µm pixel size), with 14ms between exposures. (Media 1).

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Fig. 2
Fig. 2 Schematic of X-ray phase contrast blood flow velocimetry setup. Monochromated shutter-synchronized X-ray synchrotron radiation illuminates a rotationally-symmetric vessel (in this case a 3D-printed acrylic model with constriction), through which blood is flowing in a pulsatile manner. The blood cells act as weak X-ray lenses, forming a flowing speckle pattern over the detector plane. X-ray velocimetric phase-retrieval tomography seeks to reconstruct the fully-three-dimensional time-dependent rotationally-symmetric flow velocity field within the vessel, given a time series of dynamic speckle patterns.

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Fig. 3
Fig. 3 The pulsatile flow cycle of the blood pumped through the vessel is described by (a) a full-wave rectified sine wave function, recorded over 50 image pairs, where each point represents a single pair of images. The phase-averaging (8 sections per period, i.e. A-H) of these pairs is shown in (b). Measurements here are of the axial flow component, performed using conventional PIV analysis of the images with spatial averaging over the central region of the vessel (thus indicating the modal values for that region).

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Fig. 4
Fig. 4 Geometry for velocimetric vector tomography. (a) Indicative cross-correlation map, for the z-projection of the optical thickness through the vessel, taken at two closely-spaced times. The characteristic cross-correlation peak has both horizontal (radial) and vertical (axial) components as shown. These components may be considered separately, in a tomographic velocimetry analysis. (b, c) The vertical velocity component (shown in blue) of any particle displacement is fully discernable by the X-rays which have travelled in the out-of-page (z) direction. However, a sub-component of the radial component (shown in yellow) is in the z direction and thus only the perpendicular sub-component (x-component, shown in red) contributes to the measurements.

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Fig. 5
Fig. 5 The magnitude of the Fourier transform of the speckle images showing the mask (in dark) used to suppress directional (horizontal, vertical and diagonal) artefacts, as well as low-frequency “noise”. The toroidal shape is the typical spread of frequencies comprising image speckle.

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Fig. 6
Fig. 6 Example cross correlation peaks, before (top left) and after phase retrieval (top right). Cross-correlation windows have been cropped from 256 pixels to 100 for clarity. The black spots close to the base of each peak denote the cross-correlation center or origin corresponding to zero displacement. Cross-sectional slices of both peaks are shown on the line plot (bottom center) for direct comparison. This analysis strongly suppresses the diffraction induced dark rings surrounding the correlation peaks.

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Fig. 7
Fig. 7 Reconstructed velocity profiles, for the radial (top-left), axial (top-right) components at 1.0 mm above the throat of the stenosis which has a radius of 0.95mm. A reduced susbset of the same data is shown in a combined vector plot (bottom). Measurements are from point F on the pump cycle (cf. Figure 3b).

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Fig. 8
Fig. 8 Reconstructed 2D vector velocity map, obtained using X-ray phase retrieval tomographic particle image velocimetry, for point F on the pump cycle (cf. Fig. 3b). Although 2-dimensional, it contains all the 3-dimensional flow information for a rotationally-symmetric object. The axial position is relative to the throat of the stenosis, as illustrated by the vessel schematic on the left hand side (note: measurements begin at 0.84mm above the vessel throat).

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Fig. 9
Fig. 9 Movie of reconstructed reconstructed 3+1-dimensional velocity-field vector map for pulsatile stenotic blood flow, obtained using phase-retrieval X-ray tomographic particle image velocimetry. First frame for display is point E on the pump cycle (Media 2).

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Equations (11)

Equations on this page are rendered with MathJax. Learn more.

v ( r , y ) = v r a d i a l ( r , y ) r ^ + v a x i a l ( r , y ) y ^ .
V ( x , y ) ≡ ∫ z = − ∞ ∞ v ( r , y ) . d z ,
V ( x , y ) = 2 ∫ r = x ∞ v ( r , y ) . r r 2 − x 2 . d r ,
v ( r , y ) = − 1 π ∫ x = r ∞ d V ( x , y ) d x x 2 − r 2 . d x .
v ( r , y ) = − 1 π r d d r ∫ x = r ∞ x V ( x , y ) x 2 − r 2 . d x .
v ⊥ ( r , y ) = v r a d i a l ( r , y ) cos θ ( x , y ) = v r a d i a l ( r , y ) x r ,
V r a d i a l ( x , y ) = 2 x ∫ r = x ∞ v r a d i a l ( r , y ) r 2 − x 2 . d r .
v r a d i a l ( r , y ) = − 1 π d d r ∫ x = r ∞ V r a d i a l ( x , y ) x 2 − r 2 . d x .
T ( r ⊥ , λ ) = − ( 1 μ ) ln F − 1 { ( μ / I 0 ) F [ I ( r ⊥ , λ ) ] μ + z δ | k ⊥ | 2 + ε } ,
V ( x ) = ∫ f ( s , x ) s . d s ∫ f ( s , x ) d s . T ( x ) ,
2 π ∫ r = 0 r = R v a x i a l ( r , y ) . r d r
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