Storage Phosphor System For Computed Radiography

screen optics

radiography: computed radiography: for computed Storage phosphor system system for

M. Korn D. M. Owen, and D. F. Owen, J. F. A. R. Lubinsky, J. R. Lubinsky, A. Company, Kodak Company, Eastman Kodak Research Laboratories, Eastman Rochester, Rochester, New York 14650 Abstract

to aa A model for light scattering and absorption effects is applied to is scanned with radiography in which which aa turbid turbid photostimulable photostimulable phosphor phosphor is radiography in spatial spreading of of the the laser laser light light and and the the escape escape probability of of the spatial spreading important in in determining determining system system response, response, and and models models for for their their effects effects important data. measured data. with measured compared with are compared calculations of system MTF are

computed for computed system for system The laser. The a laser. light are stimulated light are are are presented. presented. Model Model

Introduction two dimensions. scanning in two previous paper paper' reported reported some some calculations calculations on on destructive destructive scanning dimensions. A previous is, to In the the present present paper paper those those calculations calculations are are extended extended to to three three dimensions, dimensions, that is, medium. The recording medium. the recording of the depth of the depth within the place within take place that take include additional effects that of interest to us us here here is is the the calculation calculation of of depth depth effects effects in a storage phosphor/ phosphor/ interest to problem of system. scanner system. laser scanner (and of xx-rays flux of In this system2 -rays produces produces aa number number of of excited excited electrons electrons (and system an incoming flux In this long-lived in long holes), some of which are trapped -lived (storage) (storage) states states within within the the recording recording trapped in holes), luminescent produce aa luminescent to produce (scanned) to stimulated (scanned) later time the medium may be stimulated medium. At a later exposure. originalx x-ray theoriginal of the record of form aa record to form emission, which may be -ray exposure. detected to be detected emission, of the Some physically physically important important depth depth effects effects in in such such aa system are are the absorption of Some original x-ray x -ray flux flux as as aa function function of of depth, depth, the the scattering scattering and and resultant resultant spreading spreading of of the the original light. stimulating light, light, and and the the scattering scattering and and collection collection of of the the emitted light. stimulating for In the the next next sections sections we we will will generalize generalize to to three three dimensions dimensions the the integral equation for In function destructive readout readout developed developed previously previously and and present present aa solution for the transfer function destructive light stimulating light the stimulating The transfer function will depend on the spreading of the for scanning. resulting model the resulting Finally, the predictions of the light. Finally, emitted light. and on on the the escape of the emitted and data. MTF data. experimental MTF with experimental will be compared with scanning, 3 3-D for scanning, Integral equation -D equation for medium. storage medium. Let us consider the process of scanning a previously exposed storage interest. of interest. processes of illustrates the processes illustrates schematically the

x-rays

stimulation

emission

tI

x.»>

Rt

11,S

Figure 11

x y

z

exposure

scanning Figure 11

The -ray exposure exposure results results in in aa latent latent image image profile profile H(x,y,z,0), H(x,y,z,0), which which is is aa x-ray The initial x intensity Similarly, the stimulating intensity (x,y). Similarly, position (x,y). spatial position z as well as spatial function of depth z function of is z is at depth z (z), that profile I(x,y,z), I(x,y,z), as as well well as as the the probability, probability, ei0 (z), that aa photon photon created created at profile depth. of depth. functions of are functions surface, are emitted from the top surface,

(1986) IV (1986) XIV/PACS MedicineXIV 626Medicine Vol. 626 SPIEVol. 120 //SPIE /PACS IV

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Let: Let: H(x,y,z,t) H(x,y,z,t) = a measure of the latent image remaining at position (x,y,z) (x,y,z) at time t I(x,y,z,t) == the intensity of the stimulating light beam, I(x,y,z,t) beam, at at depth depth zz and and time time tt S(t)

= a a measure signal; the total light flux = measure of the output signal; flux emitted emitted from from the the top surface surface of of the the medium medium at at time time t. t. top

The cumulative cumulative exposure at depth zz and time t is The is t>

!(x,y,z,t) = E(x,y,z,t)

f I(x,y,z,t)dT, Jr ^ _««I(x,y,z,t)dr.

(1) (1)

-co

We assume, as We assume, as before, before,'1 that the reduction in the latent image, image, in aa thin layer dz, dz, is is given given in in terms terms of of the cumulative exposure: exposure: H(x,y,z,0)exp[-a»E(x,y,z,t)]dz. H(x,y,z,t)dz == H(x,y,z,0)exp[-a.E(x,y,z,t)]dz.

(2) (2)

Here "a" is ng the the "speed" Here "a" is the the parameter parameter characterizi characterizing of the storage "speed" of storage medium. The flux created at time t, t, at at depth depth z, The light light flux z, per per unit volume, is is F(x,y,z,t) = - aH(x,y,z,t) F(x,y,z,t) _ -

,

(3) (3)

at at

and and we we denote denote the the probability Probability that that aa photon photon created created at at depth depth z z escapes from from the the top top surface surface The total emitted flux is then The total

as cf(z). et (z) . as

S(t)

= fff F(x,y,z,t)ef F(x,y,z,t) et =fff

(z) (z) dxdydz.

(4) (4)

Combining Eqs. Eqs. (l)-(4), (1) -(4), we we obtain obtain SS(t) (t) == aa fff dxdydz H(x,y,z,0) fff dxdydz H(x,y,z,0) exp exp [_aft -a y* II(x,y,z,r)drJ. (x,y, z , T) di ll

L.

oo

I(x,y,z)»et( z) I(x,y,z).Ef (z)

(5) (5)

J

Equation (5) (51) is for the flux in is an an integral integral equation equation for the emitted emitted light light flux in terms terms of of the the scanning intensity strength H. intensity II and and the latent image strength H. In this equation In the the next next section we convert this equation into formula, for for the the case case of of aa rectilinear into aa convolution convolution formula, rectilinear scan. Transfer function for for scanning scanning It was for aa rectilinear It was shown shown in in ref. ref. 11 that, that, for rectilinear scan, the cumulative cumulative exposure E can scan, the can be be written as as a sum of two terms: terms: .E(x,y,z,t) == EE1(Yi-y,z) 1 (yi ~y,z) + E2(xi-x, yi-y,z) .E(x,Y,z,t)

(6) (6)

where is the resulting from all scan lines completed prior to where E-, E1 is the exposure exposure resulting to time time t, t, E 2 is the from the the current scan scan line line up up to to time time t, t, and the exposure exposure from and (xi, (x^, yi) y^) are are the coordinates of the time t. the beam center at time t. If the is taken taken as as a Gaussian whose width aa may now depend on If the scanning scanning intensity intensity profile is depth z, z, P

i - x, y i - y,z) 17,z) = 27rax,

r o o

za Y,

exp exp z

(x xi/ \ •"• ^ -- x) •"• i

L

2a 2 2a2 x,z x,z

r

2-i

2

exp -J

(Yi -- Y) y) (Yi

21 2 (7) (7)

.

202 2a 2

y,z

SPIE Medicine XIV X/V/PACS 121 SPIE Vol. Vol 626 Medicine /PACS IV /V(1986) (1986)// 121


as: defined as: P, defined pre-exposure then the preexposure functions functions R and P, R(Yi - y,z) = exp[-a.E1(yi -- y,z)] y,z)]

(8a) (8a)

y,z) == exp[-a.E2(xi - ?ç,Yi - Y,z)l, x / y i -- y,z) P(xi - x,Yi

(8b) (nb)

depth at depth G, at profile G, beam profile We define the effective beam

may be calculated by integration. direct integration. by direct as: functions as: in terms of these functions z, z, in

y,z) == a.R.P.I(xi -- x,yi x^ -- y,z) G(x i -- x,Yi G(x. -- y,z). y,z). x

(9) (9)

dz at stored charges Further, we we assume assume that that the the density density of of stored charges produced in a thin layer dz Further, layer:* the xx-ray of the attenuation of exponential attenuation the exponential to the z can be related to depth z -ray flux flux in in that that layer:* mdNx _ ray , or H(z)dz = mdN x-ray H(z)dz H(z)dz == mµNx(0)e-µzdz,, or

(10a) (lOa)

,

H(z)

(lOb) (10b)

= H(0)e-µz

(lOc) (10c)

x-ray is the x Here, Here, //,A is -ray absorption absorption coefficient, coefficient, and and mm is is the the number number of of stored stored charges charges created created absorbed xx-ray. per absorbed -ray. Using Eqs. Eqs.

Eq. with Eq. (10) , together with (9) and (10), (9) and

signal the signal for the (5) , we (5), we obtain obtain an an expression for

flux:

H(x,y,0). f dxdy H(x,y,0), S(x i ,y i ) == ff S(xi,yi) ff dxdy •*0 medium. the medium. of the thickness of is the thickness where d is profile:

G(x i -- x,^ e~/UZ e- AzG(xi x,yi -- y,z)et(z)dz

(11)

depth-weighted define aa depthWe may define weighted effective beam

d

y,z) et(z) . i - y,z)et(z) G(x i -- x,y i - y,d) G(x i - x,y x,yi G(xix,Yiy,d) = = f f dze µz G(x

(12)

0

an xx-ray absorbing an of absorbing probability of the probability to the proportional to is proportional The The weighting factor is -ray quantum at With z. With depth z. from depth photon, emitted from depth z, z, and and to to the the escape escape probability for for a luminescent photon, depth the 2 2-D and the scanning and for scanning function for spread function point spread the point G(d) defined as above, -D Fourier is the G(d) is above, G(d) G(d) scanning. for scanning. function for transfer function the transfer is the G(d) is transform G(d) light function, we spread function, find the To find the system point point spread we must must calculate the stimulating light To calculate medium. We must also calculate the medium. in the depth zz in of depth function of I(x,y,z) as a function intensity intensity profile profile I(x,y,z) problems. two problems. In the the next next two two sections we discuss two (z) . In et (z). probability et the escape probability Spreading Spreading of of the the stimulating stimulating light light of surface of I (x,y) intensity distribution light intensity (x,y) known known at the surface Given Given an an incoming incoming directed directed light distribution I (which may be in optical coefficients áa and absorption coefficients and absorption scattering and a medium with optical scattering and KK (which optical an_d Rfc and and/or layer and contact with a covering layer contact /or aa backing backing layer layer with with diffuse diffuse reflectivities reflectivities Rt ft, points?, all points at all I (IT/iF) at intensity I(r,) the intensity Rb), we would would like like to to calculate calculate the 'r, and directions 12, R, ) , we medium. within the medium. spreading the spreading to the in comparison *We *We will will neglect neglect the the spatial spatial spreading spreading of of the the stored stored charges charges in comparison to in a solid 10 keV electron in length of inelastic collision light. The The inelastgc collision length of a 10 of the stimulating light. in comparison This will will be be seen to be negligible in This example. for example. A, for 100 A, is is on on the the order of 100 the in the spreading in spatial spreading Also, we we neglect the spatial screens. Also, to to the the light light spreading in turbid screens. detector. Although the detector. in the incident xx-rays the incident by the deposited by location location of of the energy deposited -rays in (Compton incident xx-ray the incident of the spreading of substantial spreading in substantial processes processes that that can can result in -ray beam beam (Compton scattering) scattering) may may take take place, place, these these processes.are processes4are relatively relatively rare rare compared compared to photoelectric photoelectric reasonable thus, this should be aa reasonable interest, and thus, in the absorption absorption in the energy range of interest, approximation . approximation. (1986) IV (1986) XIV/PACS MedicineXIV 626Medicine Vol. 626 SPIEVol. 122 //SPIE /PACS IV


The illustrated in The problem is illustrated in Figure Figure 2. 2.

lo Io

i/ . ;.,/, ,,,./,. '„., i „ —Z-

Kr.fc) / .

/

/

1(1

I

I

L.

Figure 2 There approaches available available for for the the problem problem above. above. The first There are are two two approaches first approach approach involves involves solving radiative transfer. transfer. Since this this equation is solving the the equation equation of of radiative to is difficult difficult to solve solve except in very simple cases, usually some approximations approximations are are made. made. There are a number of except in very simple cases, usually some methods based tions that te for for particula methods based on on approxima approximations that are are appropria appropriate particular types of of problems. problems. r types The The second second approach approach is is based based on on using using aa computer computer to to simulate simulate the the processes processes of of scatterin scattering g and absorption absorption for for a a large large number number of of randomly randomly generated generated photon photon trajectories. and trajectories. This is is called the Monte Carlo Carlo method. called method. Both approaches approaches have have been been applied applied in in the the study study of of light lightscattering scatteringininx x-ray Both -ray screens. screens. this paper an approach approach based based on on radiative radiative transfer. this paper we we outline outline an transfer.

In In

Method The spatial spatial and and angular angular distribut ion of of light intensity satisfies satisfies the equation of The distribution of radiative transfer. 5 One One method of deriving deriving this this equation is is to consider a radiant energy method of radiative transfer.5 energy balance along aa particular particular beam, beam, or or "pencil "pencil", balance along ", of of radiation radiation, , as in in Figure Figure 3. 3.

I+dl I+ dI > >

I (r,SZ)

cr, K

Figure 33 5 , 6 The resulting equation equation can can be be written writtenasas::5,6 The resulting

P áX + m aY + n az

=

ß [ -I +

(ñ,St') &)I (32' ) dS2' + S (7,7i) ] f pP &

.

(13)

Here I(r,T») is intensity inin direction directionS27Fat at position positionr'; Here i(F,S2) is the the radiant radiant intensity 7; (',m,n) (/,m,n) are the the direction cosines of !3" ;; ftß is extinction coefficient coefficient (equal is the the extinction to aa ++ K), cu (equal to direction cos nes of K) , m = «/?« + K? nuff^ is single-sc attering law, Physically, term. Physicall the single -scattering law, and and SS is is the the source term. a /(a + K), p(0,ß') is the y, the the first first tern term on on the the r^ght right accounts accounts for for the the loss loss of of energy energy out out of of the the beam beam due due to to scatterin scattering or g or absorption The second term term accounts accounts for for the the gain in in_energy The second energy due to radiation from all absorption. direction s'that is scattered scattered into into the the beam beam direction direction 0. a. The The third third term represent represents the directions that is s the scatterin g of into the the beam beam from from sources sources outside outside (or scattering of direct direct radiation radiation into (or inside) inside) the the medium. medium.

SPIE Vol. SPIE Vol.626 626 Medicine Medicine XIV/PACS X/V /PACSIV IV(1986) (1986)// 123


will make two approximations. The first is to assume We will twq^approximations. assume that that the the scattering scattering law law p(ñ,ñ') p(17,!?) is is isotropic, isotropic, or or p41, p|ft,lP) = 1. 1. The second is the approximation due to Eddington,' Eddington, 7 and can be ') stated as follows. follows. F be be the the xx-component flux, and and JJ be be the the total total Let Fx component of of the the radiation flux, intensity: F(r) = f I(r,i2),Pdt2

(14a) (I4a)

J('r) f I(r',ñ)di2 J (7) == f I (7/0) dTf

((14b) I4b)

Then, from from equation equation (13), (13) , Then, Fx (x) -l/ß[a/ax / f I,Q2di1 Fx (T) == -l/£[a/ax I/ 2 dif

ax aK

Fx(x) _ -

ß

xx +

ax

aK ax

++ a/ay a/ay f/I,Qmdi2 ifmdQ

+ a/a a/a zz f/I ifndTF] + Qndt2]

aK ax xy

ay

(15a) (15a) (15b) (15b)

Xz

az

The The second second approximation approximation is is to to take take K i; . == 1/3 ijf Ki. 1/3 J8 JSi,

(16) (16)

that "mixed" integrals form KK , on the ground that that they they should should be that is, is, to to neglect neglect "mixed" integrals of of the the form small small if II is is roughly roughly isotropic. isotropic. Given Given this obtainan an equation equation for for the the total total this assumption, assumpon, weweobtain intensity: ,

222 2 2 2

2

a a i_J + 1_J 3/3 2 [(l -- wo)J o> ) J -- 477-S]. 4vrS] . 2 +áJJ 2 == 3ß2[(l dx ay az ° ax ay az

(17) (17)

We see that the effect effect of of the the approximation approximation (16) (16) has has been been to to convert convert the the integro integro-dif f-differential (13) into (17) . An equation equivalent equivalent erential equation equation (13) into the the partial partial differential differential equation equation (17). to (17) has been been used used by by Swank' Swank 8 to to discuss discussthe theMTF MTFofoflightlight-scattering screens. Swank to (17) scattering x x-ray -ray screens. considered considered aa single single line line source source of of light light inside inside the the medium medium and and calculated the Fourier In our our problem transform transform of of the the resultant resultant light light intensity intensity profile profile at the the detection plane. plane. In light source source outside outside the the medium medium (the (the laser), laser) , and and we we need need the the three threewe have a directed light solution to to Eq. Eq. (17) the following following approach. approach. dimensional solution (17).. We use the Let source function function aa single single ray ray (ß(8 -function Let us us take for the source function at at the the origin of of x,y) x,y).. We may later later find the solution for an arbitrary source distribution in in (x,y) (x,y) in in terms terms of the the 88-function -function solution. solution. S(r) = Sovj(x)$(y)e-ßz S Q 5(x)8(y)e p£t . S(r) =

(18) (18)

We may expand expand this this as: as: We may S (r) =

dkxdky e2vrixkx ex f f dkxdky

e27iyky [Soe-ßz] ey [S Qe] , ,

(19)

using 2