Chapter P Probability i Meical Imagig Cotets Itrouctio P1 Probability a isotropic emissios P2 Raioactive ecay statistics P4 Biomial coutig process P4 Half-life P5 Poisso process P6 Determiig activity of a source P6 Noise i meical imagig systems P6 Itrouctio These otes summarize a few probability topics that are relevat to meical imagig A fuametal limitatio of ay meical imagig system is the presece of oise i the measuremets There are o easy solutios to reucig oise I X-ray imagig oe coul reuce oise by icreasig the source itesity, but that woul icrease the raiatio ose to the patiet I ay moality oe coul collect multiple scas a average them to reuce oise, but this by icreases the sca time thereby icreasig the likelihoo of patiet motio artifacts (It also reuces scaer throughput, compromisig the cost effectiveess of the sca) I some moalities oe ca reuce oise by usig a etector that is more sesitive, but this comes either with a traeoff i spatial resolutio, or a icrease i the system price I ay case, oise cosieratios are very importat i meical imagig, a the basic tool use for oise aalysis is probability a raom processes Although probability is ot a official prerequisite for EECS 516, ABET requires that all accreite uergrauate egieerig programs cover probability a statistics A goo referece for the cocepts eee i this class is the book by Leo-Garcia [1] P1
P2 c J Fessler, September 12, 2007, 10:0 (stuet versio) Probability a isotropic emissios (Serves to illustrate basic raom variables cocepts) A poit source at a istace from a plaar etector emits photos isotropically (withi a plae for simplicity) The agle of emissio is a raom variable Θ Uiform[ π,π] y Poit Source θ max Θ r Y y 0 0 We wish to etermie the itesity of photos iciet o the etector, efie here as the average umber of photos per uit area (Actually, per uit legth here sice we are cosierig a 1D etector) Some etectors oly capture photos that are iciet withi a certai critical agle θ max, where 0 < θ max π/2 (For a hypothetical ieal etector we ca simply let θ max = π/2 i our fial expressios) Provie Θ θ max, the photo strikes the etector at positio Y = y 0 + ta Θ If a average of N photos are emitte from the source, the we efie the itesity o the etector to be I(y) = N p Y C (y C) P(C) where p Y C (y C) is the coitioal probability esity fuctio (pf) of Y a where C is the evet that a give photo is capture: C = [ Θ θ max ] Sice Θ has a uiform istributio over [ π,π], the capture probability is: P{C} = ( )/(2π) = θ max /π Iterpretatio of itesity: the expecte umber of photos that are recore by the etector betwee y 1 a y 2 is: y2 y 1 I(y)y From elemetary probability: p Θ (θ) = { 1 2π, θ π 0, otherwise a p Θ C (θ C) = p { 1 Θ(θ) P{C} 1 { θ θ max} =, θ θ max 0, otherwise For Θ θ max there is a mootoe ifferetiable relatioship betwee Y a Θ Thus this is a simple trasformatio of raom variables problem From elemetary trasformatio of raom variables: p Y C (y C) = p Θ C (θ C) θ y θ=ta 1 [(y y 0)/]
c J Fessler, September 12, 2007, 10:0 (stuet versio) P3 Sice /x ta 1 x = 1/(1 + x 2 ), θ y = 2 + (y y 0 ) 2 = cos2 θ = cos θ, r which ca be iterprete as a 1/r falloff of itesity (sice circumferece of circle is 2πr r) times a obliquity factor cos θ ue to agle of iciece betwee photos a etector (More o that later) Thus Alterative erivatio: p Y C (y C) = { 1 2 +(y y 0), 2 y y 0 ta θ max 0, otherwise, F Y C (y C) = P{Y y C} = P{y 0 + ta Θ y C} = P { Θ ta 1 ((y y 0 )/) C } = F Θ C (ta 1 ((y y 0 )/) C) the take erivative of both sies wrt y Mea: E[Y C] = y p Y C (y C) y = y0+ ta θ max 1 y y 0 ta θ max 2 + (y y 0 ) 2 y = y 0 where the itegral is y 0 ue to the symmetry of the pf p Y C (y C) However, for θ max = π/2 the mea is uefie (For θ max = π/2 the pf is a Cauchy istributio, for which the mea is ot efie) Variace (a measure of the sprea of the PSF): Var{Y C} = E [ (Y E[Y C]) 2 C ] = (y y 0 ) 2 p Y C (y C) y = + ta θmax = 2 1 x 2 1 2 x = (x ta 1 x) ta θ max 1 + x2 y0+ ta θ max y 0 ta θ max [(y y 0 )/] 2 1 1 1 + [(y y 0 )/] 2 y + ta θ max ta θ max = 2 where x = (y y 0 )/ a y = x The variace approaches ifiity as θ max π/2 θ max (ta θ max θ max ) Example Suppose = N = 1000 photos are emitte, let X eote the umber that strike etector betwee y 1 a y 2 Fi statistics of X The probability that a give photo strikes betwee y 1 a y 2 is p = y 2 y 1 I(y)y / = y 2 y 1 p Y C (y C) P{C} y Assumig photos are iepeet, X is a Biomial raom variable: P{X = k} = ( k where q = 1 p Mea of X is E[X] = p, variace is σ 2 X = pq ) p k q k
P4 c J Fessler, September 12, 2007, 10:0 (stuet versio) Raioactive ecay statistics (This sectio also serves to review probability: Biomial, Poisso, mea, ) For emissio tomography or uclear imagig, we will ee to kow the statistics of ecay of raioactive samples Cosier a raioactive sample cotaiig a large umber N = of (ustable) uclei at time t = 0 Let T i eote the raom variable that is the time at which the ith ucleus ecays to a stable state, eg, emits a γ photo Assume the followig The T i values are iepeet raom variables, i = 1,2,, Each T i has a expoetial istributio with mea µ T As show below, µ T is relate to the half-life of the isotope Why assume a expoetial istributio? Uique to expoetial is the memoryless property: P{T i t T i t 0 } = P{T i t t 0 } for t t 0 Also see p 349 of Leo-Garcia, 2 e for at raom property Also agrees with empirical observatios Biomial coutig process Let K(t) be raom process that couts the umber of uclei that have ecaye by time t K(0) = 0 K(t) = i=1 step(t T i), where step(t) eotes the uit step fuctio, ie, { 1, t 0 step(t) = 0, otherwise Oe expects that all uclei will ecay evetually, ie, For t 0 efie The lim P{K(t) = N = } = 1 t p t = P{T i t} = 1 e t/µt P{K(t) = k N = } = P{k out of of the T i values are less tha t} = ( k ) p k t (1 p t ) Thus K(t) is a Biomial process that couts from 0 to a icremets at raom times Note ( ) P{K(t) = N = } = p t (1 p t ) = p t = (1 e t/µt ) 1 as t The mea fuctio (versus time) is particularly importat: E[K(t) N = ] = p t = By Taylor series, 1 e x x for small x Thus for small t: E[K(t) N = ] µ T t (1 e t/µt )
c J Fessler, September 12, 2007, 10:0 (stuet versio) P5 Half-life The half-life of a isotope is the time t 1/2 for which E [ K(t 1/2 ) N = ] = 1 2 Solvig yiels: t 1/2 = µ T l 2 Alteratively, you coul efie it as the time at which the rate of ecays r t is half of iitial rate r 0 r t = t E[K(t) N = ] = t (1 e t/µt ) = /µ T e t/µt Solvig the equality r t1/2 = 1 2 r 0 yies the same expressio for t 1/2, so either iterpretatio works If is large but p t is small, the Biomial istributio is approximately same as Poisso istributio, ie, P{K(t) = k N = } e λt λk t k! where λ t = p t = (1 e t/µt ) /µ T t for t small So for t small, K(t) approximately has the first-orer istributio of a Poisso process (Derivig the kth-orer istributio is a exercise)
P6 c J Fessler, September 12, 2007, 10:0 (stuet versio) Poisso process Whe raioactive samples are purchase (eg, from a uclear reactor), the umber N of ustable uclei is ot kow exactly, but rather is a raom variable A reasoable assumptio (though this may seem circular) is that N is a Poisso raom variable with some mea µ N The by total probability the istributio of the coutig process K(t) is: ( P{K(t) = k} = P{K(t) = k N = } P{N = } = k =0 =k ) p k t (1 p t ) so uer the above assumptios, K(t) is a type of Poisso process Note that it is a ihomogeeous Poisso process sice its mea varies oliearly with time: ( ) E[K(t)] = µ N p t = µ N 1 e t/µt, ie, the rate of the process, efie as /t E[K(t)] varies with time However, for small t E[K(t)] µ N µ T t, so K(t) behaves very much like a homogeeous Poisso process for t µ T, with rate λ = µ N /µ T ( ) e µn µ N = e µn pt (µ N p t ) k /k!,! I this course we will assume, uless otherwise state, that the umber of emissios from a raioactive source i a give time iterval t is a Poisso raom variable with mea λ t, where λ is calle the activity of the source The SI uit of activity is the Becquerel (Bq), where 1 Bq is equivalet to a rate of 1 ecay (or cout ) per seco I aily practice i the US, the historical uit of Curies (Ci) is still use frequetly, where 1 Ci = 37 10 10 Bq Determiig activity of a source Suppose you have a source (such as mouse liver extracte from a mouse previously ijecte with a raiotracer) where λ is ukow If we cout the source i well couter for time t 0 ; we observe K(t 0 ) (total umber of couts i t 0 secos) The maximumlikelihoo estimator for λ is: ˆλ = K(t 0) t 0 This estimator is ubiase because: ] [ ] K(t0 ) E[ˆλ = E = E[K(t 0)] = λt 0 = λ t 0 t 0 } Var{ˆλ = Var t 0 { } K(t0 ) t 0 = Var{K(t 0)} t 2 0 = λt 0 t 2 0 As t 0 icreases, variace of ˆλ ecreases, ie, the precisio of the estimate improves = λ t 0 Noise i meical imagig systems From above, imagig with raioisotopes yiels Poisso measuremets X-ray imagig also is omiate by Poisso oise The emissio of photos from the X-ray source is very realistically moele by a Poisso process sice the curret to the aoe esures a cotiual supply of photos, i cotrast to uclear imagig where there is oly a fiite (but large) umber of uclei i a give source The omiat oise source i both NMR a ultrasou systems is electroic oise This ca be simply moele as a zero-mea, Gaussia process Aother oise source i ultrasou (a optical a RADAR systems) is speckle oise This oise is irectly relate to the coherece of ultrasou imagig Bibliography [1] A Leo-Garcia Probability a raom processes for electrical egieerig Aiso-Wesley, New York, 2 eitio, 1994