Critical absorption tomography of small samples

Nuclear Instruments and Methods 206 (1983) 547-552 North-Holland Publishing Company

CRITICAL

ABSORPTION

TOMOGRAPHY

547

OF SMALL

SAMPLES

Proposed applications of synchrotron radiation to computerized tomography I1 L. G R O D Z I N S

Physics Department, Massachusetts Institute of Technology, Cambridge, Mass., USA Received 18 March 1982 and revised form 12 August 1982.

Computerized X-ray transmission tomography can be used to reconstruct the distribution of a particular element in a section by comparing the image obtained at a photon energy just greater than a critical absorption edge with that obtained at a lower photon energy. The method is examined, with particular regard to samples smaller than l cm, to determine the sensitivity to element concentration as a function of spatial resolution, sample size and material, and photon flux. With the intense, monochromatic and tunable X-ray sources that are becoming available from dedicated synchrotron radiation facilities, it will be practical to make three-dimensional maps, with spatial resolution in the micrometer range, of trace elements in the interior of small specimens.

1. Introduction Computerized X-ray t o m o g r a p h y is now a routine diagnostic m e t h o d for reconstructing the distribution of linear a t t e n u a t i o n coefficients in the cross section of a sample. In a c o m p a n i o n paper [1] we examined the optimization of the parameters of computerized X-ray transmission t o m o g r a p h y (hereafter abbreviated as CTT) w h e n small samples are examined. This paper is an analogous study of C T T applied to critical edge absorption measurements to yield the distribution of atomic elements in the cross section of small samples. C T T yields the distribution of the linear a t t e n u a t i o n coefficients, t~, of the reconstructed section. If the absorption is d o m i n a t e d by the C o m p t o n effect, then the distribution of /~ values is a distribution of electron densities. If the p h o t o n energy is low enough so that the photoelectric effect is the d o m i n a n t c o m p o n e n t of the total interaction cross section, then the/~ values reflect, in a non-simple way, the elemental composition in the picture elements (pixels). These well-known facts [2] can be exploited to give a measure of the heavy element c o m p o n e n t in the pixels by c o m p a r i n g C T T scans taken at two energies, one of which is low enough to that the/L values contain a substantial c o m p o n e n t of the photoelectric cross sections [3]. The c o n c e n t r a t i o n of a specific element can be obtained by making use of the sharp increase in the absorption coefficient as the p h o t o n energy exceeds the b i n d i n g energy of a b o u n d electron of that element. The comparison is then made between the reconstructed images obtained with two p h o t o n energies that closely bracket such a so-called critical a b s o r p t i o n edge [4]. The straightforward method of making this comparison is to take successive tomo0 1 6 7 - 5 0 8 7 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 0 3 . 0 0 © 1983 N o r t h - H o l l a n d

graphic scans, one at an energy below the critical edge, a n o t h e r at a n energy above. A n alternative is to scan once with a b e a m whose energy spectrum includes the critical edge, and sort the transmitted b e a m with an energy or wave length dispersive detector system. In either case, well-collimated beams are needed with intense c o m p o n e n t s at energies a r o u n d the critical edges of the sought for elements. Only the heavier elements ( Z >_ 50) can be studied in living h u m a n subjects since the lower energies required for lighter elements are excessively absorbed. Even iodine, considered by Riederer a n d Mistretta for critical-edge tomographic studies of h u m a n s [4], presents severe difficulties since the intensity of the 33 keV radiation, needed for K-edge absorption, is attenuated by a factor of a t h o u s a n d on traversing only 20 cm of tissue. For lighter elements, a useful transmission signal in h u m a n subjects can only be obtained with incident fluxes so large as to exceed the patient's tolerance limit for radiation. For this reason the thrust of this p a p e r is not directed to the s c a n n i n g of elements for clinical studies of h u m a n patients. As a correlary to the preceding remarks, it follows that the lighter the element to be investigated, the smaller must be the size of the sample; e.g., 10 keV radiation, needed for the study of zinc, is attenuated by a factor of a t h o u s a n d on traversing just 1.4 cm of water. As the sample size gets smaller, the b e a m flux must be increased if the sensitivity to the elements and the relative spatial resolution are to be maintained. The relation between p h o t o n energy a n d sample size is considered in ref. 1. A n ideal p h o t o n source for critical absorption t o m o g r a p h y should provide high intensity beams of nearly monoenergetic p h o t o n s whose mean energy can

548

L. Grodzins / Critical absorption tomography

be tuned across the critical edges of the elements. The beams from synchrotron radiation accelerators have precisely these characteristics. The effective X-ray densities from dedicated facilities are a million times greater than from conventional X-ray machines. A n d the photon beams can be focussed, m o n o c h r o m a t i z e d and tuned. The use of such beams should make computerized critical absorption tomography (CCAT) a sensitive, non-destructive analytical probe of elemental distributions in the interior of small samples of interest to m a n y disciplines.

2. Fluence requirements It is the purpose of this paper to examine the limits to the potential of C C A T for obtaining the quantitative distribution of elements as a function of spatial resolution, sample size and material, and p h o t o n flux. The analytic procedure used here is essentially that of refs. 5 and 1. The geometry is that shown in figs. la and l b of ref. 1. For simplicity, the geometrical section to be scanned is a disc-shaped slice of diameter D and thickness h, out of a long cylinder. A single b e a m of cross section, wh, is used to obtain a pixel resolution element of area w 2 and volume w2h. The grid parameter, m = D / w , is the n u m b e r of pixels of dimension w traversed by a ray passing along a diameter. A scan consists of ~m2/4 i n d e p e n d e n t measurements taken in the usual way; i.e., the sample is moved with respect to the b e a m in m linear steps of length w for each of ~rm/4 incremental rotations of 4 / m rad. The flux of detected p h o t o n s of energy Ey is N0e UTD, where /~T is the mean total linear a t t e n u a t i o n coefficient and N o is the incident flux. Let N be the n u m b e r of photons of energy Ey which, during a full scan of one section, traverse a given pixel. The standard deviation of N is a measure of the minim u m observable change due to a change in ff in one pixel. In one pixel, AN = NwAff so

N '/2 = N( WffT ) ( A # / p . T ) .

or, Np t~f---- ~--( Om + % f ) = l~m(l + fOc/Om), Iooo

. . . .

'

(3)

.....

,oo WATE MEDIUM

7


!

/ i

i,o

/ j ~ K Zn

Pb K

co

(1)

A/~ is the difference between the total attenuation coefficient with and without the presence of critical atoms in the pixel. Experimentally, that difference is obtained by measuring the absorption at energies just above and below the critical edge. The assumption that one standard deviation suffices for ascertaining the limit is unwarranted in m a n y cases. The more stringent requirem e n t of 3o introduces a further factor of 9 in the equations below. Expressing eq. (1) in terms of the total incident p h o t o n s in a full scan taken in time T,

DB e ~rD N ° w h T - w(WffT)2(Aff/ffT) 2 "

The factor D / w comes from the fact that in a full scan, each of the rn 2 separate a t t e n u a t i o n measurements involves the traversal of m pixels. The factor B accounts. as shown by Chesler et al. [5], for the b r o a d e n i n g of the r e s o l u t i o n / s e n s i t i v i t y due to the reconstruction algorithm. For the best algorithms in present use, B -- 2 [5]; we take B = 2 in all calculations. [Eq. (2) ignores several factors which tend to cancel: a factor of ~ / 4 in the total n u m b e r of pixels; another factor of ~ / 4 in the mean p a t h length through the disc of diameter D; the compensating increase of about a factor of 2 in time since two measurements are needed, one above and one below the critical edge.] The thrust of this paper is to determine the lower limits of elemental composition which can be determined by CCAT. In such cases, A# << # f and eq. (2) can be written in a more convenient form. Expressing # v in terms of the cross sections for the p h o t o n interaction with the medium, on,, and with the critical atoms,

(2)

I.O

~~K AI

IRON MEDIUM K Ca Zn K

0

. . . . .

:

'

'

,

IO P H O T O N ENERGY, keV

i

,

i

i

10o

Fig. 1. The ratio of the total interaction cross section, for a photon of energy E, on an atom of atomic number Z, to the total interaction cross section for a molecule of water (upper, heavy curves) and to an atom of iron (lower dotted curves). The ratios are confined to the region around the K edges for the elements Z considered, except for the ratio for Pb vs. water which is given over the range encompassing the L edges as well.

L. Grodzins / Critical absorption tomography where N is Avogadro's number, A is the mean atomic weight of the medium, p is its density and f is the fraction of the atoms which are assumed to be critical; f < < 1. Substituting eq. (3) into eq. (2), and dropping the subscript m,

NowhT -

DB e #D w( w~ )2( foc/° )2

.

(4)

A governing parameter is oc/o; the larger it is the smaller is f. The quotient is plotted in fig. 1 for a number of elements in water and iron media [6,7]. A number of aspects of fig. 1 are worth noting: For a given medium, the ratio pc/o, and, therefore, f, is a strong function of E v. When o decreases with energy faster than does pc, the ratio continues to increase above the critical energy. The fractional change in p i p across a critical K edge is independent of the medium and gradually decreases with Z, from a factor of 12 at aluminum to a factor of 4 at lead; a factor of 4 should be adequate to render a minimum concentration of critical atoms invisible to photons of energy just below the critical edge. The L edges are more effective than the K edges for the heaviest elements in a water-like medium. The illustration of Pb in water in fig. 1 also shows that only on crossing the lowest member of an atomic shell, in this case the L3 edge, is the change in oc large enough for unambiguous identification of trace elements. The ratio % / 0 is consistently lower for the iron matrix than for the water matrix; as expected, the heavier the matrix the less sensitive is the technique. The j u m p across a critical edge has a relative maxim u m of 8.5 for Mn in Fe, vanishes for Fe in Fe and remains below 5 until Br in Fe.

3. Discussion To illustrate the behavior of f as a function of Z c sample size and medium, a few special cases are considered. These are meant to be illustrative only since investigators will generally have their own special conditions on one or several of the variables. We consider first an optimization procedure for CCAT. In ref. 1, the CTT analogue of eq. (4) was minimized to find the optimum value of/~ and hence the optimum E v so as to obtain the greatest sensitivity for a given total flux. That value, ~t = 2/D, can be readily obtained if tunable, monochromatic X-ray beams are available since /~ is an exponential function of Ey over a broad range of /z. The situation is quite different for C C A T where the photon energy is fixed by the critical edge of the sought-for atoms and thus the energy dependent variables, p., pc and o are also fixed. If D and w are unrelated, then there is no minimum, apart from the

549

trivial case of D = w, in which case there is no tomography. On the other hand, if D / w is constant, then eq. (4) can be optimized. Setting tt = k / D , eq. (4) becomes

UowhT( fOc/Ct)2

ek

(D/w)3B

k2

(5)

The lefthand side of eq. (5) is minimized for k = 2, The resulting optimized equation is

NowhT= 2B( D / w ) 3 (foe~O)2 '

(6)

which corresponds to eq. (5) of ref. 1. Again, the total n u m b e r of p h o t o n s per scan d e t e r m i n e s the resolution/sensitivity product, ( w / D )3( foc/o ) 2. Keeping in mind that the optimization procedure is normally impractical since it requires that the sample size be varied by orders of magnitude if a broad span of elements is scanned, it is nonetheless instructive to see the limits obtainable with a given fluence. We take for illustration the specifications of the examples of ref. 4, i.e., 4 × 101° incident photons per scan, m = D / w = 100, and a sensitivity to/~ of 1% in each pixel. [The choice of a fluence of 4 × 10 *° photons is not wholly arbitrary since it represents the nominal fluence needed in CAT scanning of human subjects to obtain electron densities to an accuracy of 1% in pixels of (1 mm) 2 resolution. Since that fluence should also be obtainable at resolutions of (1 ~m) 2 in practical times using synchrotron beams, it is a convenient bench mark for comparative calculations.] In fig. 2, f versus Z is plotted for a water medium and both K and L3 edge absorption are shown. In fig. 3, the media are silicon and iron; only K edge absorption is considered since for no value of Z is L edge absorption as sensitive. The photon energies for critical absorption are given along the abscissae. In water-like specimens it is possible to determine heavier element concentrations ( Z > 40) down to 10-20 atoms per million atoms of the host, in each of 104 pixels using 4 × 101° incident photons of appropriate energy. (The specimen sizes will be greater than about 10 cm, and E v >_. 15 keV.) For lower Z elements, the sensitivity falls rapidly; by a factor of 10 from Z = 40 to Z = 20, and another factor of 10 on dropping to Z = 13. As expected, the sensitivity to trace elements worsens as the matrix gets heavier. For the elements above lead, a silicon matrix yields sensitivities almost equivalent to water hut an iron medium reduces the sensitivity by an order of magnitude. In the region around Z = 50, the sensitivity has deteriorated to 10 -4 for silicon and l0 ~3 for iron matrixes. At the self-critical edges, the sensitivity is 1%. Just below the edges the sensitivity increases again by about an order of magnitude. The luxury of choosing the optimum size sample to obtain the minimum trace sensitivity is unattainable or impractical for most situations. It is likely that most

L. Grodzins / Critical absorption tomography

550

CRITICAL

ABSORPTION

TOMOGRAPHY

WATER MEDIUM 4u I0 I° INCIDENT PHOTONS PER SCAN SPATIAL RESOLUTION • Io-zD D IS OPTIMUM FOR EACH ENERGY

10"2 CO

o ..J

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90

z Fig. 2. The minimum fraction of atoms of atomic number Z which can be detected in a water environment by CCAT when the sample size is optimized for each critically absorbed photon energy. The spatial resolution is 1% of the specimen diameter and 4× 10 l° incident photons of appropriate energy are used per scan.

studies will be directed to the determination of the distributions of one or many elements in a given sample at some specified level of spatial resolution. To illustrate the sensitivity of C C A T in such situations we have used eq. (4) to calculate the minimum fraction f of elements which can be determined in water-like samples having diameters of 100 #m, shown in fig. 4, and 1 m m and 1 cm diameters, shown in fig. 5. In all cases, the fluence is NowhT= 10 l° photons and m = 100; the photon beam densities in the three cases are 4 × 10 Is, 4 × 1016 and 4 × 1014/cm 2 respectively. The minima of the K edge curves, given in figs. 4 and 5, follow closely the envelope of the " o p t i m u m " condition of fig. 2. Away from the " o p t i m u m " element which can be studied in a sample of a given size, the sensitivity worsens: below the " o p t i m u m " , the deterioration is due to the exponential absorption which reduces the number of photons counted; above the optimum, the sensitivity worsens because the interaction per pixel gets smaller. For the 100/Lm specimen, fig. 4, one wants to keep Ey close to 3 keV [1]. As heavier and heavier elements are investigated, the choice switches from K edge to L3 edge to M5 edge absorption when these energies reach the neighborhood of 3 keV. In this way, the sensitivity can be kept in the range of 100 1000 p p m for each ( # m ) 2 pixel, f o r 4 × 101° incident photons, and m = 100.

1

r

T

CRITICAL

CRITICAL

CO

o ._1 <~

ABSORPTION

TOMOGRAPHY

SILICON AND IRON MEDIA K EDGE ABSORPTION 4xlO IO INCIDENT PHOTONS PER SCAN SPATIAL RESOLUTION " I0- 2D D IS OPTIMUM FOR EACH ENERGY 10 -2

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WATER MEDIUM 4=IOI°INCIDENT PHOTONS PER SCAN D= O,Olcm , w = I p r n

CO

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T

ABSORPTION TOMOGRAPHY

z

10"4

Z

I0 -S I0 l

40

9~7

~

18

±

29

4~4

~

61

L

83

E KeY 5- K,

IO -3

I0 20

30 4 0 5 0 6 0

70 8 0 9 0

z Fig. 3. Similar to fig. 2 but the media are silicon and iron.

20

30

40

50

60

70

80

90

IO0

Z

Fig. 4. The minimum atomic fraction of critical atoms of atomic number Z that can be detected by CCAT in a 100 #m diameter water-like specimen. The resolution is 1 #m and there are 4× 101° incident photons per scan.

L. Grodzins / Critical absorption tomography CRITICAL ,

O3

o I-

ABSORPTION

LL

SCAN

10"2!

._1
TOMOGRAPHY

WATER MEDIUM 4 ' ¢ I O ' ° I N C I D E N T PHOTONS PER SPATIAL RESOLUTION • IO'~D D= O.Icrn ond Icm

~

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i

z

o_ EDGE

,

0

10-4"

_

n~ LL

~S

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I0 20 30 40 50 60 70 80 90

Z Fig. 5. Similar to fig. 4. The specimens are l mrn and 1 cm in diameter. The spatial resolution is 1% of the diameter and there are 4 x 10 l° incident photons per scan.

Analogous remarks apply to the results for 1 mm and 1 cm specimens, though in these cases the M5 edges are no longer useful because M5 energies are never in the optimum range. The study of light elements can only be done with small samples and the heavier the matrix the smaller the samples must be. In a water-like medium, C C A T is insensitive to minor constituents below about phosphorous in 100/Lm samples, below about potassium in 1 mm samples and below about iron in 1 cm samples; for much larger samples only K edge absorption need be considered and only the heavier elements can be studied. To study such light elements as Na and Mg in tissue, the specimens should be smaller than - 30 ffm.

4. Synchrotron X-ray sources The smaller the desired spatial resolution, for a fixed grid size, the greater must be the flux density and hence the greater the radiation dose to the specimen. In the illustrations above, the parameters - F = 4 x 10 l°, m = 100, and Jlx/l~ = 1% - are typical of those for C A T scanning of humans. In such cases, D > 15 cm and the flux density required for a few-second scan using a single detector approach the maximum obtainable from conventional X-ray sources. And the radiation dose

551

approaches the maximum permissible to the patient. When the same parameters, and the same scan time, are assumed in the study of lighter elements, under the optimum conditions of eq. (6), the needed flux density increases inversely as the square of the resolution; the radiation dose increases more rapidly [1]. If the size is not optimized for the element studied, then the flux density, and the radiation dose, increase even faster with decreasing spatial resolution. We conclude from these simple arguments that C C A T studies of elements in the lower part of the periodic table will not be feasible using conventional X-ray sources and that the smaller specimens studied by this technique must be tolerant of high radiation doses. At this time only synchrotron light sources can provide the flux densities of tunable, homogeneous beams needed for C C A T of small samples. Happily, there are a growing number of dedicated synchrotron radiation facilities throughout the world which provide X-ray beams well-matched to the requirements of C C A T [8]. For example, the National Synchrotron Light Source (NSLS) at Brookhaven National Laboratory, which will soon be coming on line, is expected to provide X-ray fluxes of 1015 p h o t o n s / s . m r a d . A E / E = 1%, at energies below about 25 keV, with lower fluxes at higher photon energies [9]. These primary beams can be focussed and monochromatized; design calculations for the NSLS facility [10] indicate that focussed flux densities of 10 l° p h o t o n s / ( f f m ) 2 at A E / E = O . I % , can be obtained at Er < 10 keV, dropping by a factor of 2 at 20 keV and by another factor of 5 at 40 keV. Such flux densities will make computerized critical absorption tomography feasible at the parts per million elemental sensitivity at resolutions below about 10 ffm for a wide range of elements in the lighter matrixes. To stress this claim, consider the sensitivity to trace elements obtained in a 90 min scan using the above design fluxes of focussed synchrotron beams. In water-like specimens of 100 ffm size investigated at a resolution of (1 ffm) 3, the sensitivity to elements from phosphorous to bromine would be below 50 ppm. At the minima, see fig. 4, the sensitivities would be about 20 ppm for calcium, 10 ppm for tin and 10 ppm for uranium. If the spatial resolution increased to (4 ffm) 3, then the sensitivities could be improved by a factor of 32 [see eq. (4)] bringing them into the partsper-million range. For 1 mm diameter water-like samples investigated at a resolution of (10 ffm) 3, the sensitivity to elements from iron through bromine can be below 5 ppm. Finally, we note that computerized tomography can be applied to the fluorescent X-rays. (In that case, the exciting radiation can be charged particles, as long as multiple scattering is negligible.) This complementary technique for determining elemental concentrations will be discussed in a following paper. In summary, we find that computerized fluorescence tomography has, corn-

552

L. Grodzins / Critical absorption tomograpl~v

pared with computerized critical absorption tomography, strong advantages: the sensitivity appears linearly rather than quadratically in the equations analogous to eqs. (2) and (4) of this paper, the requirements on the energy homogeneity of the incident b e a m are not so severe, b e a m intensities may be higher, several elements can be studied simultaneously and, in the reconstruction algorithm, one will contend with but one array of measurements rather than the subtraction of two arrays of large, nearly equal numbers. There are also strong disadvantages: fluorescent yields are small for the lighter elements, one c a n n o t use a simple current measuring detector as in C C A T but must count individual p h o t o n s with good energy resolution and reasonably high efficiency, and backgrounds under the photopeaks are a critical problem especially with large grid sizes. The two techniques are competitive. The choice of which to use will depend on a variety of factors including the degree of inhomogeneity in the distribution of the elements u n d e r study. In m a n y cases it will be possible and desirable to do the fluorescence and the critical absorption scans simultaneously. 1 am pleased to acknowledge the hospitality of the Niels Bohr Institute where this work began.

References [1] L. Grodzins, this issue, preceding article. [2] R.A. Brooks, LG. Mitchell, C.M. O'Connor and G. Di Chiro, Phys. Med. Biol. 26 (1981) 141. [3] R.A. Rutherford, B.R. Pullan and I. lsherwood, Neuroradiology 11 (1976)23. A.J. Talbert, R.A. Brooks and D.G. Morganthaler, Phys. Med. Biol. 25 (1980) 261. [4] S.J. Riederer and C.A. Mistretta, Med. Phys. 4 (1977) 474. [5] D.A. Chesler, S.J. Riederer and N.J. Pelc, J. Comput. Assist. Tomogr. 1 (1977) 64. [6] Photon cross sections, except for water, were taken from E. Storm and H.I. Israel, Nucl. Data Tables 7 (1970) 565. [7] The data for a water medium were taken from J.H. Hubbell, Rad. Res. 70 91977) 58. [8] See, for example, Proc. Japanese/USA Seminar on Synchrotron radiation facilities, Nucl. Instr. and Meth. 177 (1980). [9] A. Van Steenberger, Nucl. Instr. and Meth. 172 91980) 25. [10] C.S. Sparks Jr., in Synchrotron radiation research, eds., H. Winick and S. Deniach (Plenum, New York, 1980) pp. 459-512.