4: From x-ray to image Computed Tomography

Transcription

1 4: From -ra to imag Computd Tomograph 1. What factors influnc contrast in -ra imaging? Bam hardning Snsitivit and rsolution considrations 2. What influncs CNR of -ra imaging? 3. What is th fundamntal basis for imag rconstruction using -ra absorption? Radon Transform 4. How can -ra imags b rconstructd? Sinogram Backprojction vs. filtrd backprojction Cntral Slic Thorm 5. Eampls & Summar Aftr this cours ou 1. Undrstand th consquncs of th Brmsstrahlung continuum on imag contrast 2. Undrstand how Compton scattring rducs imag contrast and how its influnc can b rducd 3. Ar familiar with th Radon transform 4. Undrstand th principl of matri rconstruction and backprojction 5. Undrstand th major mchanisms lading to CT contrast What dos absorption in th ral world impl? Linar attnuation cofficint n n-dn Contrast is wll-dfind for monochromatic -ras d If μ is constant in : linar attnuation cofficint Unit: [cm -1 ] n( ) N Th masurmnt that is wantd: (,) What is masurd: n() But, μ=f(e,z, ) Two consquncs: Bam hardning Dpth dpndnt contrast n( ) ln N ( for a homognous objct of thicknss ) 4-4

2 What dos th Enrg Spctrum of an -ra tub rall look lik? filtrd Brmsstrahlung and charactristic mission i(e ) E ff Intraction with orbital lctrons I = i(e )de i(e ): compl function Brmsstrahlung continuum Dfin ffctiv photon nrg E ff Soft -ras (low nrg) Hard -ras (high nrg) Maimal -ra nrg is = kintic nrg of - (V cathod ) Minimal nrg is zro, but: Soft -ras (low nrg) ar filtrd b instrumnt E ff is incrasd b instrumnt (filtring of soft -ras) 4-5 What is th consqunc of nrg-dpndnt absorption? Bam Hardning - Effctiv nrg dpnds on dpth A similar consqunc ariss in tissu: Idal: Monochromatic -ras (E () = ( )) Ralit: Polchromatic, multinrgtic i(e ) Absorption is not uniform with E Contrast changs with larg objcts and dpth Ecssiv radiation dos to suprficial tissu Solution : Rduc i(e ) for soft -ras (.g. 3mm Al liminats 9% of 2kV photons) 4-6

3 4-2. How dos -ra scattring impact CNR? Objct (lsion) Scattring incrass with FOV (fild-of-viw) of irradiation Furthr rduction of imag contrast Photolctric ffct: complt absorption of photons Objct (lsion) asil dtctd With E Compton scattring incrasd masking of objct. Solution: Anti-scattr grid (collimator) Slic slction Collimation principl: stablishs dirctionalit of -ras Dtctor High Z, : stop -ras 4-7 How is CNR quantifid? Signal: I(d) (no. photons dtctd) I (no. photons irradiatd) CT intnsit can b masurd in absolut trms (CT-numbr) Soft tissu: Tpicall has wak contrast (small Hounsfild units) Contrast: I(d) du to μ(d) diffrncs i(e ), C producs rducd contrast Compton scattring: Antiscattr grid HU : attnuation normalizd to watr () rang from -1 (air) to +3 (bon and contrast agnts) soft tissus: -3 to +1 CT numbr watr watr 1 4-8

4 4-3. What is th basis of imag rconstruction? Th Radon transform I() I( I( I( Givn a crtain bam intnsit (no. of photons) I at a givn position, I( ), th bam intnsit at + is I( 2) I( ) 2) I( 2) I( ) ) I( ) ) ( ) ( ( ) Rcursiv application to driv I( +2) ( ) ( ) ) ( ) Considring a two-dimnsional objct: I I dt ctd Radon transform g() Dfinition Radon transform of a point-lik homognous objct I dt ctor (, ) d sourc I : intnsit of incidnt -ra bam g ( ) (, ') d' rctangular objct lim I I( dt ctd ) dt ctor ( ') d' sourc g() 4-9 Dos ach pil hav a uniqu trajctor? Sinogram Radon transform of a circular objct Radon transform = projction of objct Trajctor: Rsin(+) Dtctor: is movd in circular motion around objct (indicatd b angular position ) Each point in spac is uniqul rprsntd b Amplitud R and phas of sinusoidal trajctor in Sinogram (sic!): (,) ) 4-1

5 Can a CT imag b constructd b Matri invrsion? Dcomposing an objct into a 22 matri rquirs a minimum of 4 masurmnts: 12 I I 1 ln(i 1 /I ) = ln(i 2 /I ) = ln(i 3 /I ) = ln(i 4 /I ) = Stting = ilds a linar 22 invrsion problm linking k to I k In principl such an n 2 matri can b invrtd. Too compl, computationall intnsiv and unstabl CT was introducd in 197 simpl rconstruction algorithm! What algorithm is adaptd to 197 s computing powr? Backprojction rconstruction Basic rconstruction principl: Along th masurd projction dirction fill in ach pil constant numbrs corrsponding to th Radon transform (projction intnsit). Rpat for nt orintation of th projction, sum th valus in ovrlapping pils. Illustration with gra shads (point-lik objct): 2 Projctions 4 Projctions R f cos sin, sin cos cos sin sin cos d 8 Projctions 16 Projctions 4-12

6 Wh dos simpl Backprojction hav poor spatial rsolution? Backprojction has poor spatial rsolution: Rconstruction of a pointobjct falls of with 1/r WHY? Th rconstruction falls off with 1/r (in analog to th dcras of light intnsit in 2D) dd=da=sin(d)ddd d da d d d Numbr of ras (projctions): constant with d But: pil siz = dd da = const No. of ras 1/ 4-13 How can good imag rsolution b maintaind? Filtrd Backprojction 4-14

7 How is Backprojction linkd to Fourir transform? Cntral Slic Thorm Imag spac FT k-spac (Fourir spac) k Final Imag (CT: acquird data) (, ) IFT M ( k, k ) M (, k ) 2i ( k k ) (, ) dd (, ) d 2ik d k = FT of projction -spac Rconstruction using FT : artifacts k ( k, k ) (, ) ik ik dd MRI: Acquird Data S also: Signals and Sstms (SV) X-ra CT : Eampls (Human) Bon (calcification) : bright (high absorption) Imaging of mummifid bodis Diffrnt dnsitis of tissu giv intr-mdiat rsults Air is dark Dislodgd arrow had 4-16

8 CT: Eampls (mous) Pr- Post-contrast 3D CT scan of rodnt spin tratd with human msnchmal stm clls (transducd with th human BMP-9 gn via an adnoviral vctor) significant bon formation at th tratmnt sits (arrows) 13μm micro CT of mous placnta vasculatur Micro-CT of mous fmor bon 4-17 CT: Summar Main contrast is bon vs. soft tissu (or air) (calcium contnt i.. - dnsit ) Contrast agnts (incras Z ff ) allow dpiction of vssl architctur and lsions SNR and CNR: 1. Intnsit can b incrasd b cathod currnt 2. High spatial rsolution possibl (limitd onl b radiation dos in humans) 4-18

9 How hav CT scannrs volvd? Gnrations of CT scannrs First Gnration Paralll bam dsign On/two dtctors Translation/rotation 2nd Gnration Small fan bam Translation/rotation Largr no. of dtctors 3rd Gnration Multipl dtctors Larg fan bam 4th Gnration Dtctor ring Larg fan bam 4-19