Mathematics for Computed Tomography

Intensity measurements Scanning Patient X-Ray beam X-Ray detector Intensity measurements Computer Memory

Scanning X-ray tube & detectors rotate around patient (All recent scanners) Detectors measure radiation transmitted through patient for various pencil beam projections Relative transmission calculated Fraction of beam exiting patient Patient X-Ray beams

CT Detectors electronic / quantitative extremely sensitive small radiation input differences measureable output digitized & sent to computer

Photon Phate What can happen to an x-ray photon passing through a material (tissue)? Material ??? Incoming X-ray Photon

Photon Phate #1: Nothing Photon exits unaffected same energy same direction Good These photons form the CT image Material Incoming X-ray Photon Outgoing X-ray Photon

Photon Phate #2: Absorption Photon disappears Its energy is absorbed by material Good Creates differential absorption which forms CT image Bad Source of patient dose Material Incoming X-ray Photon

Photon Phate #3: Scatter Lower energy photon emerges energy difference deposited in material Photon usually emerges in different direction Bad Degrades image Material Outgoing X-ray Photon Incoming X-ray Photon

Photon Beam Attenuation Anything which removes original photon from beam absorption scatter Material Incoming X-ray Photon Absorption Material Incoming X-ray Photon Outgoing X-ray Scatter

Example Beam Attenuation (Mono-energy source) Each cm of material reduces beam intensity 20% exiting beam intensity 80% of incident for 1 cm absorber 1cm 1cm 1cm 1cm 100 100 * .8 = 80 80 * .8 = 64 64 * .8 = 51 51 * .8 = 41

Attenuation Equation for Mono-energetic Photon Beams I = Ioe-mx For photons which are neither absorbed nor scattered I = Exiting beam intensity Io = Incident beam intensity e = constant (2.718…) m = linear attenuation coefficient property of absorber material beam energy x = absorber thickness Material I Io x

More Realistic CT Example Beam Attenuation for non-uniform Material 4 different materials 4 different attenuation coefficients x #1 ? #2 ? #3 ? #4 ? Io I m1 m2 m3 m4 I = Ioe-(m1+m2+m3+m4)x

Effect of Beam Energy on Mono-energetic Beam Attenuation Low energy photons more easily absorbed High energy photons more penetrating All materials attenuate a larger fraction of low than high energy photons Material Material 80 <80 100 100 Higher-energy mono-energetic beam Lower-energy mono-energetic beam

Attenuation Coefficient & Beam Energy m depends on beam energy as well as material x #1 ? #2 ? #3 ? #4 ? Io I m1 m2 m3 m4 I = Ioe-mx I = Ioe-(m1+m2+m3+m4)x

Mono-energetic X-ray Beams Available from radionuclide sources Not used in CT Radionuclide intensity much lower than that of x-ray tube

Mono-energetic beam equation! X-ray Tube Beam x High intensity Produces poly-energetic beam Characteristic radiation Bremsstrahlung #1 #2 #3 #4 Io I m1 m2 m3 m4 Mono-energetic beam equation! I = Ioe-(m1+m2+m3+m4)x

Beam Hardening Complication Beam quality changes as it travels through absorber greater fraction of low-energy photons removed from beam Average beam energy increases 1cm 1cm 1cm 1cm A B C D E Fewer Photons but kVavg(B) > kVavg(A) Fewer Photons but kVavg(C) > kVavg(B) Fewer Photons but kVavg(D) > kVavg(C) Fewer Photons but kVavg(E) > kVavg(D)

Beam Hardening Complication Attenuation coefficients mn depend on beam energy!!! Beam spectrum incident on each block unknown Four m’s, each for a different & unknown energy 1cm 1cm 1cm 1cm m1 m2 m3 m4 I = Ioe-(m1+m2+m3+m4)x

I = Ioe-(m1+m2+m3+m4 +...)x Reconstruction Scanner measures “I” for thousands of pencil beam projections Computer calculates tens of thousands of attenuation coefficients one for each pixel Computer must correct for beam hardening effect of increase in average beam energy from one side of projection to other I = Ioe-(m1+m2+m3+m4 +...)x

Why is CT done with High kV’s? Less dependence of attenuation coefficient on photon energy Attenuation coefficient changes less at higher kV’s High kV provides high radiation flux at detector

Image Reconstruction IA = Ioe-(mA1+mA2+mA3+mA4 +...)x One of these equations for every projection line IA = Ioe-(mA1+mA2+mA3+mA4 +...)x Projection #A Projection #B IB = Ioe-(mB1+mB2+mB3+mB4 +...)x IC = Ioe-(mC1+mC2+mC3+mC4 +...)x Projection #C

Reconstruction Calculates: Image Reconstruction * The equations IA, IB, IC, ... What We Measure: Projection #A IA = Ioe-(mA1+mA2+mA3+mA4 +...)x mA1, mA2, mA3, ... Reconstruction Calculates: mB1, mB2, mB3, ... mC1, mC2, mC3, ... Etc. Projection #B IB = Ioe-(mB1+mB2+mB3+mB4 +...)x Projection #C IC = Ioe-(mC1+mC2+mC3+mC4 +...)x

CT (Hounsfield) Number Calculated from reconstructed pixel attenuation coefficient (mt - mW) CT # = 1000 X ------------ mW Where: ut = linear attenuation coefficient for tissue in pixel uW = linear attenuation coefficient for water

CT Numbers for Special Stuff Bone: +1000 Water: 0 Air: -1000 (mt - mW) CT # = 1000 X ------------ mW