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Phase Contrast in X-Ray Imaging

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Phase Contrast in X-Ray Imaging Phase contrast in x-ray imaging A thesis submitted for the degree of Doctor of Philosophy by Anthony Wayne Greaves Centre for Atom Optics and Ultrafast Spectroscopy Faculty of Engineering and Industrial Sciences Swinburne University of Technology Melbourne, Australia 2010 for Lucy and Kalinda \ In every kind of magnitude there is a degree or sort to which our sense is proportioned, the perception and knowledge of which is of the greatest use to mankind. The same is the groundwork of philosophy; for, though all sorts and degrees are equally the object of philosophical speculation, yet it is from those which are proportioned to sense that a philosopher must set out in his inquiries, ascending or descending afterwards as his pursuits may require. He does well indeed to take his views from many points of sight, and supply the defects of sense by a well-regulated imagination; nor is he to be con¯ned by any limit in space or time; but, as his knowledge of Nature is founded on the observation of sensible things, he must begin with these, and must often return to them to examine his progress by them. Here is his secure hold; and as he sets out from thence, so if he likewise trace not often his steps backwards with caution, he will be in hazard of losing his way in the labyrinths of Nature," | (MaC'laurin: An Account of Sir I. Newton's Philosophical Discoveries. Written 1728; second edition, 1750; S( p. 18,19.) Good judgment comes from experience; experience comes from bad judgment. | Mark Twain Declaration I, Anthony Wayne Greaves, declare that this thesis entitled: \Phase contrast in x-ray imaging" is my own work and has not been submitted previously, in whole or in part, in respect of any other academic award. Anthony Wayne Greaves Centre for Atom Optics and Ultrafast Spectroscopy Faculty of Engineering and Industrial Sciences Swinburne University of Technology Australia Dated this day, April 6, 2011 Abstract X-ray phase contrast imaging (PCI) has the potential, over certain energy ranges, to improve conventional x-ray diagnostic practice by utilizing the higher phase than absorption coe±cient in the complex index of refraction for low atomic Z materials, thereby potentially enhancing the di®erences between healthy and cancerous tissue. Mammography is a particularly suitable application due to its high biopsy incidence. Of the di®erent types of PCI, the in-line holographic method or propagation-based imaging (PBI) has the ability to utilize polychromatic x-ray sources. In clinical practice, polychromatic x-ray sources require ¯ltering to remove the lower energy rays that have insu±cient energy to penetrate the body and provide useful diagnostic information. However, the introduction of ¯lters in PBI is known to cause phase contrast degradation, and this thesis examines both experimentally and theoretically, the various factors a®ecting phase contrast. Experimental factors using a Feinfocus micro-focus, tungsten x-ray source with 0.5 mm beryllium inherent ¯ltration, included: varying the tube voltage and source size, introducing extra ¯lters of di®erent materials and thicknesses including grain size and surface smoothness, and objects of di®erent shapes and materials and at various positions in the beam. Image detection was via Fuji image plates and BAS- 5000 scanner. The most convenient ¯lter material was aluminium, and the object showing the best phase contrast was a thin walled, cylindrical ¯bre (Cuprophan RC55) normally used in dialysis. The x-ray spectrum of the tube was also measured using a i Abstract ii SiLi detector system, with and without various ¯lters. Theoretically, phase contrast was simulated via geometric ray optics, Fresnel and Fresnel-Kirchho® wave di®raction methods. Inelastic scattering was also included in a simple model but was found to have only a very minor e®ect. Simulations showed good qualitative concurrence between the ray optics and Fresnel di®raction methods, with the Fresnel method providing greater detail in the line pro¯le shape of the object, but essentially showing the same behaviour. Phase contrast reduction was found to be caused mainly by beam hardening, which narrows the phase peaks obtained from dispersion in the object, and together with source and detector considerations which then smooth the narrower ¯ltered phase peaks more than the un¯ltered case. There were some discrepancies with theoretical simulations producing absolute values approximately twice that of experimental ones, but the overall behaviour of phase contrast with the imaging parameters was consistent. Acknowledgements Acknowledgements to Peter Cadusch for his help with analysis and simulation. To Andrew Pogany for his help with theory and discussion. To Andrew Stevenson for help with experimental work and equipment. To David Parry for help in constructing PMMA step wedges and stages to hold ¯bre test objects. To Kevin O'Donnell for preparing the various grain sizes in the Aluminium ¯lters, and to Peter Lynch for help with the MCA. Thanks to Peter Miller for the use of his MCA program and Sherry Mayo for suggesting imaging ¯bres and James Kelly for help with some of the illustrations. iii Contents Declaration Contents iv List of Figures ix List of Tables xxvi 1 Introduction 1 1.1 Foundation . 3 1.2 Experimental . 4 1.3 Theoretical and simulations . 5 1.4 Discussion and Conclusion . 6 2 Historical Introduction 7 2.1 Phase . 7 2.1.1 X-ray di®raction . 9 2.2 Phase contrast . 12 2.2.1 Holography . 13 2.2.2 Phase and coherence . 15 2.3 X-ray phase contrast . 17 2.3.1 Interferometry . 17 2.3.2 Di®raction Enhanced Imaging . 18 iv CONTENTS v 2.3.3 In Line Holography . 20 2.4 Some Applications . 25 2.4.1 Phase Retrieval . 26 2.4.2 Medical Applications . 29 2.5 Summary . 31 3 Phase contrast theory 33 3.1 Ray tracing and geometric optics . 34 3.1.1 Projection to a screen and the Jacobian . 36 3.1.2 Cylindrical phase object . 39 3.2 Wave optical approach . 41 3.2.1 Fresnel-Kirchho® Equation . 42 3.2.2 Rayleigh Sommerfeld Integrals . 44 3.2.3 Fraunhofer and Fresnel di®raction . 46 3.3 Fourier Optics . 52 3.3.1 Angular spectrum . 54 3.3.2 Propagation of Angular Spectrum . 55 3.3.3 Fresnel Approximation and Angular Spectrum . 56 3.4 Transmission function . 57 3.4.1 Phase edge . 58 3.4.2 Cylindrically shaped phase objects . 60 3.4.3 Source and detector considerations . 62 3.4.4 Summary . 66 4 Interaction of x-rays with matter 68 4.1 Attenuation, absorption and extinction . 69 4.1.1 Absorption . 72 4.2 Scattering . 75 CONTENTS vi 4.2.1 Classical scattering . 77 4.2.2 Dipole radiation . 78 4.2.3 Scattering by a free electron - Thomson scattering . 83 4.2.4 Scattering by a single bound electron - Rayleigh scattering . 86 4.2.5 Scattering by a multi-electron atom . 87 4.2.6 Atomic scattering factors . 89 4.3 Refractive index . 91 4.4 Inelastic scattering . 94 4.5 Scatter modelling . 95 4.5.1 A simple model of x-ray scatter . 98 4.5.2 Evaluation of scatter . 103 4.5.3 Summary . 105 5 Experimental preliminaries 107 5.1 Equipment . 108 5.1.1 X-ray source . 108 5.1.2 X-ray detection . 109 5.1.3 Scanner resolution . 111 5.1.4 Characteristic curve . 112 5.1.5 Image reproducibility . 113 5.2 Test objects . 117 5.2.1 Step wedge . 117 5.2.2 Phase contrast variation with distance . 119 5.2.3 E®ect of ¯lter position . 121 5.2.4 Finer PMMA step wedge . 123 5.2.5 Other edges . 123 5.2.6 Fibres . 133 5.2.7 Dialysis ¯bres . 133 CONTENTS vii 5.3 Fibre straightening . 139 5.4 Scatter measurements . 142 5.4.1 Scatter variation with air gap . 149 5.4.2 Scatter variation with thickness . 153 5.4.3 Scatter variation with ¯lter position . 157 5.5 Summary . 162 6 Experimental 164 6.1 Phase contrast variation with source size . 165 6.2 Phase contrast variation with tube potential . 169 6.3 Phase contrast variation with source to object distance . 172 6.4 Phase contrast variation with ¯lters . 177 6.4.1 Aluminium grain size . 182 6.4.2 Errors . 185 6.4.3 Filters and noise . 187 6.5 X-ray spectrum . 192 6.6 Summary . 199 7 Refraction 206 7.1 Ray tracing . 207 7.1.1 Analytical approach . 207 7.2 Intensity pro¯le . 216 7.3 Energy spectra . 222 7.4 Cellulose . 223 7.4.1 Combinations of refractive materials . 227 7.5 Summary . 229 8 Di®raction simulations 232 8.1 Fresnel-Kirchho® formulation . 234 8.1.1 Fresnel-Kirchho® formulation in 1 dimension . 234 8.1.2 Incident plane wave on an edge . 236 8.1.3 Spherical Waves . 240 8.1.4 Fresnel-Kirchho® equation for a long straight cylinder . 242 8.1.5 Direct integration of Fresnel-Kirchho® equation . 245 8.1.6 Fresnel zones . 246 8.2 Fresnel-Kirchho® for smaller ¯bres . 250 8.3 Fresnel formulation . 256 8.3.1 One dimensional mask approximation . ..
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