X-Ray Notes, Part I. Images are characterized by the interaction of x-ray photons and tissue.

Transcription

1 oll 4 X-ray oes : Page X-Ray oes, Par I X-ray Imaging Images are characerized by he ineracion of -ray hoons and issue. Physics Definiion: Radiaion a sream of aricles or hoons. Paricles: α + He, e - elecrons, β elecrons emied from nuclei, β + osirons, + roon, n neurons Phoons: -ray, γ, annihilaion hoons, ec. Models for ineracion of radiaion and maer:. Absorion generally low kineic energy KE. Scaering 3. o a yical ineracion a gradual loss of energy he charged aricles above α, e -, β, β +, + inerac very srongly wih issue and yically do no ass comleely hrough he human body and hus canno be used for imaging. Of he above aricles hoons and neuronsn ass hrough he body wih an aroriae amoun of ineracion for imaging oo lile is also bad. Behavior of Radiaion Along a Line Assumions:

2 oll 4 X-ray oes : Page. Maer consiss of discree aricles searaed by disances ha are large comared o he size of he aricles.. For a given ah lengh along a line, an -ray hoon eiher ineracs wih rob. or i doesn and all ineracions are indeenden. 3. Scaered hoons scaer a a differen angle and don conribue o he coninuing flu of hoons along he line. he change in he number of hoons is: d d d µ d d d µ e µ ' d' were µ is he linear aenuaion coefficien and has unis disance -. For a consan µ: he Basic X-ray Imaging Sysem e µ ow consider a arallel ray -ray flu ha has inensiy I inensiy is hoons/uni area/uni ime he asses hrough a 3D objec having a disribuion of aenuaion coefficiens µ,y,z and rojecs o an image I d,y:

3 oll 4 X-ray oes : Page 3, y, z I d, y I e µ dz Generaion of -rays - arge is usually a high-z, heavy elemen yically W, ungsen. - Elecrons are acceleraed by he volage beween he cahode and he anode. - A oenial energy of Eqv e.g. e * 5 kv 5 kev all ges convered o kiniic energy E ½ m e v e.g. also 5 kev. Kinds of elecron ineracions: a. Inelasic energy absorbing scaering wih aomic elecrons he ejecion of a bound elecron followed by emission of a hoons from sonaneous energy sae ransiions. he Bohr model accouns for absorion/generaion of discree valued energies kev is one characerisic -ray for W. Any combinaion of shell ransiion energies will also be characerisic energies e.g. 3. and 6.7 kev. Very low energies are hard o observe due o oher absorion rocesses.

4 oll 4 X-ray oes : Page 4 b. Bremssrahlung Braking Radiaion Acceleraion change in direcion of elecron by Coulomb aracion o he large, osiively charged nucleus leads o he generaion of hoons acceleraion of any charged aricle will do his. For elecrons of a aricular energy, E, sriking an infiniely hin arge, Bremssrahlung radiaion will have a uniform disribuion of energy beween and E. We assume ha all elecrons inerac. For a hick arge, i is ofen modeled as a series of hin arges where he highes energy iminging uon subsequen sages is reduced by he ineracions. Each hin arge roduces a new uniform secrum, bu wih a lower eak energy. he resulan secrum is aroimaely linear from a eak a kev o a E.

5 oll 4 X-ray oes : Page 5 he -ray Secrum - For elecrons wih energy E, he maimum -ray hoon energy is E. hc - E hυ - Very low energy hoons are absorbed by he arge and by he glass in he -ray ube. - Secrum will have a combinaion of Bohr discree energies and Bremssrahlung radiaion: - he -ray secrum is funcion of hoon energy: I I E - I now reresens energy/uni ime/uni area or ower/uni area. Pracical -ray ube Why ungsen? - -ray secrum in desired range - High Z high efficiency in soing elecrons - High meling oin 33 deg. C yical oeraion em is ~5 deg. C his is due o he low efficiency of he elecron o -ray conversion ~.8%. he res goes ino hea. - Eamle: - Roaion of arge o reduce eak em

6 oll 4 X-ray oes : Page 6 - Shielding o colae beam - Window furher filers -ray secrum hardens beam makes i have a higher average E he Aenuaion Coefficien We say above ha he -ray secrum is a funcion of hoon energy E: I I E. he aenuaion funcion is also a funcion of E: µ µ,y,z,e. he new eression for he inensiy a he ouu will no be: I, y I E e µ, y, z, E dzde d E oe: I d ells us nohing abou z or E i only gives us,y informaion. he -ray aenuaion coefficien µ is, of course, also a funcion of maerial roeries. wo of he mos imoran roeries ha affec he aenuaion coefficien are issue densiy, ρ, and he aomic number Z. As mos -ray hoon/issue ineracions are hoon/elecron ineracions boh ρ and Z will influence µ. For -ray hoons, here are 4 main yes of ineracions lised in order of increasing likelihood wih increasing hoon energy, E:. Rayleigh-homson Scaering. Phooelecric Absorion 3. Comon Scaering 4. Pair Producion In general, we can wrie an eression for he aenuaion coefficien as he some of hese consiuen ars: µ E µ E + µ E + µ E + E + r e cs µ. Rayleigh-homson Scaering or coheren scaering aomic absorion wih sonaneous emission a he same energy E. his is he same effec as is seen in -ray

7 oll 4 X-ray oes : Page 7 diffracion in crysals. his erm is rarely imoran in he diagnosic energy range 5- kev.. Phooelecric Absorion Absorion of hoon o ionize and ejec an aomic elecron. he ejeced elecron will have an kineic energy of he hoon energy less he binding energy of he elecron. he hooelecric effec increases raidly wih aomic number, Z, and wih decreasing energy. he hooelecric effec dominaes µ in he lower ar of he diagnosic secrum. For high Z maerials e.g. Lead, Iodine, ungsen, he shell energy boundaries are eviden in he µ vs. E los. When he energy ges high enough o make ha shell s elecrons available o he PE effec when E eceeds he binding energy, hen he robabiliy of a PE ineracion increases. 3. Comon Scaering scaing of hoons by an elasic collision wih a free elecron. Elasic collisions reserve E and momenum. For loosely bound elecrons or very high energy hoons, he equaions for free elecrons hold reasonably well.

8 oll 4 X-ray oes : Page 8 Unknowns: φ, θ, E, K.E. Conservaion of energy: where m m K.E. E E m m c is he relaivisic mass of he elecron v / c Jus a check on his equaion for v << c, hen m m c m m v c c v Conservaion of momenum in and y direcions: E c E' sinθ mv sinφ c / c mv E' cosθ + mv cosφ c c. solving hese equaions we ge he energy of he scaered hoon: E ' + E E e E cosθ where E e m c 5 kev, he res energy of an elecron. Commens: - For E << E e, here is very lile change in energy wih angle. - For higher E:

9 oll 4 X-ray oes : Page 9 - For low E, scaer is essenially isoroic in angle - For higher E, scaer is referenially forward scaered where here is very lile change in hoon E. - I is very hard o discriminae beween forward scaered hoons and unimeded hoons based on energy. - µ cs is nearly consan across diagnosic secrum - Comon scaer comes mosly from aomic elecrons µ cs is roorional o ρ - A higher E, Comon scaer dominaes over he PE effec mos imoran effec in -ray imaging. 4. Pair Producion he sonaneous creaion of an elecron/osiron air: In his ineracion, hoon energy in ransferred o mass energy in he elecron and osiron. Since he res energy of each is 5 kev, air roducion canno occur for -ray hoons below kev no in he diagnosic secrum. Posirons will wander around unil hey bum ino an elecron, which will resul in muual annihilaion and he emission of wo 5 kev hoons: he ejeced hoons from a osiron/elecron annihilaion is he basis for osiron emission omograhy [more on his laer].

10 oll 4 X-ray oes : Page oal Linear aenuaion coefficien for hoons Again, he combined coefficien is: µ E µ E + µ E + µ E + E + r e cs µ For eamle, he combined coefficien for lead is: An alernae o linear aenuaion coefficien is he mass aenuaion coefficien which is defined as: τ µ / ρ unis: cm /gm his arameer is convenien when describing he behavior of comosie maerials wih consiuen comonens: τ m i M i where m i are he masses of he comonens and M is he oal mass. τ i Beam Hardening Because he aenuaion secrum is no uniform across he diagnosic energy secrum, he ouu secrum will have a differen inensiy disribuion han he inu secrum, I E. If we sli an objec ino several smaller ars, and look a hen energy secrum a for each ar:

11 oll 4 X-ray oes : Page we will find ha he mean energy: E EI E de I E de will increase ge harder as we move hrough he objec: E < < E < E <... E n. For medical imaging, his has he unforunae consequence ha a aricular issue ye will have a µ ha changes as a funcion of osiion along he ah. In aricular, as we move deeer ino he objec, we will find ha here is less aenuaion han eeced, given he iniial secrum, I E. One soluion is o make he beam hard o begin wih. his is ofen accomlished by filering ou he low E hoons wih a hin meal lae ofen use aluminum. Comon Scaered X-rays Consider he following objec wih an -ray oaque core: he ouu image migh look like his:

12 oll 4 X-ray oes : Page on which we can define a conras C S / S and a conras o noise raio CR S / σ. ow, consider ha he scaered hoons here some fracion of he s scaered hoons will scaer forward and will generae addiional hoons in he final image. he disribuion of he scaered hoons will look somehing like he objec convolved wih he forward scaering disribuion. he final image will be he sum of he ransmied hoons and he scaered hoons. By increasing S and σ s he scaered hoons will reduce boh he conras and he conras o noise raio. How many hoons are scaered? Derived from Macovski, Problem 3.4 Le s look a an objec of lengh l having an aenuaion coefficien µ µ + µ +. Le be r e µ cs he number hoons inciden uon he objec and ha he number of hoons ha have no ineraced a deh is. he number of scaered hoons in an inerval d will be:

13 oll 4 X-ray oes : Page 3 cs µ cs d and he oal number of scaered hoons will be: cs l cs d l l µ d cs µ e µ d cs µ cs e µ l µ oe ha -e-µl is he oal number of hoons ha inerac wih he objec. oise in X-ray Sysems In an -ray sysem, images yically are creaed from inensiy values ha are relaed o he number of hoons ha srike a deecor elemen in a finie eriod of ime. he hoons are generaed by elecrons randomly sriking a source and hus he hoons a he deecor are also random in naure. We yically describe his kind of random rocess as one having a rae arameer, unis: evens/ime, and an observaion ime,. Le X be he random variable R.V. ha describes he number of evens hoons sriking he deecor elemen in ime. X will be a Poisson disribued random variable wih arameer. E.g. X ~ Poisson Derivaion of Poisson Disribuion Below, we will derive he Poisson disribuion from a se of indeenden Bernoulli R.V. s. Le be some small ime inerval and / be he number of indeenden rials. he robabiliy of an even hoon in inerval will be. Each Bernoulli rail will hen be an R.V.:

14 oll 4 X-ray oes : Page 4 Y i ~ Bernoulli Y even A, wih robabiliy i even B, wih robabiliy q We also assume ha is chosen o be small enough so ha he robabiliy ha here are wo evens is very small laer we will le go o zero, so his is a non-issue. ow we consider he sum of he evens, which yields a binomial R.V. X Y i i X ~ Binomial, he robabiliy densiy funcion is f Probabiliy{X } he robabiliy ha here were evens in ime. For a binomial R.V., his is derived from he following: A M A B M B of even ye A of even ye B which will occur wih robabiliy q and here are o ge of even ye A. his yields he following.d.f.: Please also observe ha f!!! q! differen ways!! he mean of X is: f

15 oll 4 X-ray oes : Page 5 y f q y y - y ' q q X E X y y y y ' ' ' '! '! '! and, and leing,!!!!!! ] [ In a similar fashion we can show ha [ ] q X E X + and, σ Finally, we will le, /,, and q. In he following, kee in mind ha q,, /. he Poisson robabiliy disribuion is herefore: [ ] [] [][ ]!!!!!!! / e e q q q q f q + + L L [he eonenial i comes from ε ε ε ε / e e.] he mean and variance are: q X X σ

16 oll 4 X-ray oes : Page 6 Here X is a Poisson R.V. wih arameer : X ~ Poisson. SR of a Poisson Measuremen In general, he iel values in an -ray image are disribued according o a Poisson R.V. If he mean value of he hoon couns for a iel is µ, hen he signal o noise raio of for ha iel will be: X SR σ X µ µ µ he SR increases as he square roo of he number of hoons. hus, he SR increases as he square roo of he dose o he aien. Finally, by averaging ogeher wo neighboring iels, we can roughly double he hoon couns and imrove he SR by. he above figure shows Poisson disribuions as he mean increases from 3 o 5. We can see ha he disribuion becomes more symmeric and Gaussian.

17 oll 4 X-ray oes : Page 7 he above figure akes Poisson disribuions and normalizes hem by heir mean, ha is, we subrac he mean and divide he -ais by he mean. his lo show demonsraes ha he widh of he disribuion as a fracion of he mean. As he mean ges larger, he disribuion ges roorionaely narrower he sd. dev. vs. mean raio is smaller SR is higher. he Relaionshi of a Poisson Process o he Eonenial R.V. Le be an eonenial R.V. ha describes he ime beween evens in a Poisson rocess. he derivaion follows. Recall ha he robabiliy ha an even occurs in inerval will be. Also, noe ha he robabiliy ha no even occurs in inerval will be q -. ow, suose he we wan o know wha is he robabiliy ha no even occurred beween and. his is he same as saying ha we have / inverval in which no even can occur. If hese inervals are indeenden ha is saying ha he hoons don inerac wih each oher or end o come in grous or whaever, he robabiliy ha no even occurred beween and will be q : / { no even occurs in, } Pr we again deermine his funcion as : Pr / { no even occurs in, } [ ] e

18 oll 4 X-ray oes : Page 8 he robabiliy densiy funcion of, f, describes he robabiliy ha an even occurs a ime and robabiliy disribuion funcion of inegral of f describes he robabiliy ha an even occurs by ime will be equal o: e, for F Pr{ no even occurs in, }, for < and he robabiliy densiy funcion is he derivaive of his funcion: e, for f, for < he eonenial R.V. is a coninuous R.V. of he imes beween evens and is described as: which has a mean an variance of: ~ Eonenial σ Memoryless Proery he eonenial R.V. is memoryless, meaning ha disribuion and densiy of even imes in he fuure is no affeced by as evens, ha is, a any oin in ime, he ime unil he ne even is an eonenial R.V. wih arameer. his is he same as saying ha jus because we haven seen an even in a long ime, we re no more likely o have an even soon. Jus like he gambler s fallacy. Secifically, { > + > } Pr{ > } Pr which says given ha an even hasn occurred by ime, he robabiliy ha an even will no occur by ime + will be same as he robabiliy ha no even occurs in,. Proof:

19 oll 4 X-ray oes : Page 9 { } { } F F F > + + > + > Pr e e e Pr