1!TT Liu, BE280A, UCSD Fall 2010!Bioengineering 280A"Principles of Biomedical Imaging"Fall Quarter 2010"X-Rays Lecture 1"TT Liu, BE280A, UCSD Fall 2010!TT Liu, BE280A, UCSD Fall 2010!EM spectrum!Suetens 2002!TT Liu, BE280A, UCSD Fall 2010!X-Ray Tube!Suetens 2002!Tungsten filament heated to about 2200 C leading to thermionic emission of electrons. !Usually tungsten is used for anode!Molybdenum for mammography!2!TT Liu, BE280A, UCSD Fall 2010!X-Ray Production!Prince and Links 2005!Collisional transfers!Radiative transfers!TT Liu, BE280A, UCSD Fall 2010!X-Ray Spectrum!Prince and Links 2005!Lower energy photons are absorbed by anode, tube, and other filters!TT Liu, BE280A, UCSD Fall 2010!Interaction with Matter!Photoelectric effect dominates at low x-ray energies and high atomic numbers. !Typical energy range for diagnostic x-rays is below 200 keV.!The two most important types of interaction are photoelectric absorption and Compton scattering.!Compton scattering dominates at high x-ray energies and low atomic numbers, not much contrast !http://www.eee.ntu.ac.uk/research/vision/asobania! € Ee−= ν− EBTT Liu, BE280A, UCSD Fall 2010!X-Ray Imaging Chain!Suetens 2002!Reduces effects of Compton scattering!3!TT Liu, BE280A, UCSD Fall 2010!X-ray film!Flexible base!~ 150 µm!Emulsion with!silver halide crystals!Each layer!~ 10 µm!Silver halide crystals absorb optical energy. After development, crystals that have absorbed enough energy are converted to metallic silver and look dark on the screen. Thus, parts in the object that attenuate the x-rays will look brighter. !TT Liu, BE280A, UCSD Fall 2010!Intensifying Screen!http://learntech.uwe.ac.uk/radiography/RScience/imaging_principles_d/diagimage11.htm!http://www.sunnybrook.utoronto.ca:8080/~selenium/xray.html#Film!TT Liu, BE280A, UCSD Fall 2010!X-Ray Examples!Suetens 2002!TT Liu, BE280A, UCSD Fall 2010!X-Ray w/ Contrast Agents!Suetens 2002!Angiogram using an iodine-based contrast agent. !K-edge of iodine is 33.2 keV!Barium Sulfate!K-edge of Barium is 37.4 keV!4!TT Liu, BE280A, UCSD Fall 2010!Intensity!€ I = EφEnergy!Photon flux rate!€ φ=NAΔtUnit Time!Unit Area!Number of photons!TT Liu, BE280A, UCSD Fall 2010!Intensity!€ φ= S(ʹ′ E 0∞∫)dʹ′ E X-ray spectrum!€ I = S(ʹ′ E 0∞∫)ʹ′ E dʹ′ E TT Liu, BE280A, UCSD Fall 2010!Attenuation!€ Iout= Iinexp(−µd)d!For single-energy x-rays passing through a homogenous object: !Linear attenuation coefficient!TT Liu, BE280A, UCSD Fall 2010!Attenuation!€ n =µNΔx photons lost per unit lengthµ=n /NΔx fraction of photons lost per unit length€ ΔN = −n€ dNdx= −µN€ N(x) = N0e−µx€ I(Δx) = I0e−µΔxFor mono-energetic case, intensity is!5!TT Liu, BE280A, UCSD Fall 2010!Attenuation!€ dNdx= −µ(x)N€ N(x) = N0exp −µʹ′ x ( )0x∫dʹ′ x ( )Inhomogeneous Slab !€ I(x) = I0exp −µʹ′ x ( )0x∫dʹ′ x ( )Attenuation depends on energy, so also need to integrate over energies!€ I(x) = S0ʹ′ E ( )ʹ′ E 0∞∫exp −µʹ′ x ;ʹ′ E ( )0x∫dʹ′ x ( )dʹ′ E TT Liu, BE280A, UCSD Fall 2010!Attenuation!5 10 50 100 150 1 0.1 Attenuation Coefficient 500 Bone Muscle Fat Adapted from www.cis.rit.edu/class/simg215/xrays.ppt Photon Energy (keV) Photoelectric effect dominates!Compton Scattering dominates!More Attenuation!Less Attenuation!TT Liu, BE280A, UCSD Fall 2010!Half Value Layer!Values from Webb 2003 X-ray energy (keV)!HVL, muscle (cm)!HVL Bone (cm)!30! 1.8! 0.4!50! 3.0! 1.2!100! 3.9! 2.3!150! 4.5! 2.8!In chest radiography, about 90% of x-rays are absorbed by body. Average energy from a tungsten source is 68 keV. However, many lower energy beams are absorbed by tissue, so average energy is higher. This is referred to as beam-hardening, and reduces the contrast. !TT Liu, BE280A, UCSD Fall 2010!Contrast!Bushberg et al 2001!6!TT Liu, BE280A, UCSD Fall 2010!Contrast!TT Liu, BE280A, UCSD Fall 2010!€ A = N0exp(−µx)B = N0exp(−µ(x + z))CS=A − BA=N0exp(−µx) − N0exp(−µ(x + z))N0exp(−µx)=1− exp(−µz)Subject Contrast!Bushberg et al 2001!TT Liu, BE280A, UCSD Fall 2010!X-Ray Imaging Geometry! Prince and Links 2005!TT Liu, BE280A, UCSD Fall 2010!Inverse Square Law! Prince and Links 2005!€ Inverse Square LawI0=IS4πd2Id(x, y) =IS4πr2 where r2= x2+ y2+ d2=I0d2r2= I0cos2θ7!TT Liu, BE280A, UCSD Fall 2010!Obliquity Factor! Prince and Links 2005!€ Obliquity FactorId(x, y) = I0cosθa!€ a /cosθa!TT Liu, BE280A, UCSD Fall 2010!X-Ray Imaging Geometry!€ Beam Divergence and Flat PanelIr= I0cos3θExample : Chest x - ray at 2 yards with 14x17 inch film.Question : What is the smallest ratio IrI0 across the film?€ rd= 72+ 8.52= 11cosθ=drd2+ d2= 0.989IrI0= cos3θ= 0.966TT Liu, BE280A, UCSD Fall 2010!Anode Heel Effect!http://www.animalinsides.com/radphys/chapters/Lect2.pdf!TT Liu, BE280A, UCSD Fall 2010!Compensation Filters! Prince and Links 2005!8!TT Liu, BE280A, UCSD Fall 2010!Path Length! Prince and Links 2005!€ ʹ′ L = L /cosθId(x, y) = I0cos3θexp(−µL /cosθ)L!L’!€ θTT Liu, BE280A, UCSD Fall 2010!Magnification of Object!€ M(z) =dz=Source to Image Distance (SID)Source to Object Distance (SOD)Bushberg et al 2001!z!d!TT Liu, BE280A, UCSD Fall 2010!Magnification of Object!M = 1: I(x,y) = t(x,y)!M = 2: I(x,y) = t(x/2,y/2)!In general, I(x,y) = t(x/M(z),y/M(z))!t(x,y)! I(x,y)!I(x,y)!TT Liu, BE280A, UCSD Fall 2010!X-Ray Imaging Equation! Prince and Links 2005!€ At z = d there is no magnification, soId(x, y) = I0cos3θ⋅ exp −µ(s)dsLx,y∫cosθ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ = I0cos3θ⋅ td(x, y)where tz(x, y) is the transmittivity of the object at distance z€ In general, with magnificationId(x, y) = I0cos3θ⋅ tz(x M(z), y M(z))9!TT Liu, BE280A, UCSD Fall 2010! Prince and Link 2005!TT Liu, BE280A, UCSD Fall 2010!Source magnification!€ m(z) = −d − zz= −BA= 1− M(z)Bushberg et al 2001!d!=z!€ DimageDfocal=d − zzTT Liu, BE280A, UCSD Fall 2010!Image of a point object!€ Id(x, y) = ks(x / m, y /m)ks x m(z), y m(z)( )∫∫dxdy = constant⇒ k =1m2(z)€ Id(x, y) = limm →0s(x /m,y /m)m2=δ(x, y)s(x,y)!TT Liu, BE280A, UCSD Fall 2010!Image of arbitrary object!€ limm →0Id(x, y) = t(x, y)s(x,y)!s(x,y)!€ Id(x, y) = ???m=1!€ Id(x, y) =cos3θ4πd2m2s(x /m, y / m) **t(x / M, y / M)t(x,y)!t(x,y)!10!TT Liu, BE280A, UCSD Fall 2010!Convolution!TT Liu, BE280A, UCSD Fall 2010!Macovski 1983!M=1!m=0!M=2!m=-1!TT Liu, BE280A, UCSD Fall
View Full Document
Unlocking...