A Brief Description

Tables of differential and total elastic scattering cross sections for 93 elements in the range Z=13 to Z=104 and for selected photon energies in the range 50 to 1500 keV have been published. The values of the cross sections have been obtained by accurate interpolation from the published cross section values available for ten elements and seven photon energies obtained using the state-of-the-art precise S-matrix method. The S-matrix method is known to produce Rayleigh scattering amplitudes in most cases accurate to the order of 1%. The elastic scattering cross sections presented in this table includes all elastic scattering processes, namely Rayleigh scattering by bound electrons, nuclear Thomson scattering and Delbruck scattering. This table contains elastic scattering cross section values for all elements for 14 photon energies (50.0, 59.54, 145.4, 316.5, 411.8, 468.1, 661.6, 778.9, 964.0, 1004.8, 1112.1, 1274.5, 1408.0, 1500 keV) for 55 different scattering angles in the range 0<theta<180 degrees. Angles were chosen judiciously, very close angular grids (namely, 14 angles between 0 and 1 degree) due to the sharp variation of cross sections at forward angles and at relatively coarser grids with increasing scattering angles. The energies are chosen to cover the commonly used photon energies and in a suitable grid so as to enable one to use the usual interpolation methods to determine the cross sections at other energies. We will also be happy to comply with any request from scientists to provide cross section values at any customized scattering angles and energies. Angle integrated total elastic scattering cross sections at all energies and elements are also presented in the table.

Comparison of experimental cross sections with the tabulated values is also made and presented graphically. Comparison shows that the predicted cross sections are, in general, in good agreement with the measurements. Although this comparison is not exhaustive, it gives a representative example about its suitability for use by experimentalists and radiation physicists. The paper also contains a table depicting the number of experimental measurements performed at each energy and element. This table along with the figures indicates the region of photon energy and elements for which further measurements are needed.

Elastic scattering is an important mode of interaction of photons with matter. The tables of precise elastic scattering cross sections for any element and for varying photon energies have long been needed, not only from the point of view of fundamental science, but also from the point of view of their applications in various fields such as material study, health physics, biology, medicine, etc. The authors will be happy if the radiation physicists find this table useful.

Introduction

The increased porosity of bone, as a consequence of the reduction of bone mineral density (BMD), is known generally as osteoporosis. BMD loss may be localised to certain bones (or areas of bone), perhaps resulting from the disuse of a limb, or it may involve the entire skeleton due to the manifestation of a metabolic bone disease. Although the whole skeleton may be affected, osteoporosis manifests itself in certain regions to a greater extent than others, especially in regions with a greater surface area of bone.

Bone may be classified as either cortical bone or trabecular bone. The surrounding structure of bone is a hard, dense shell, known as cortical bone. Regions of bone that are filled with marrow are hollow except for a series of bony struts known as trabecular bone. These trabeculae enclose the bone marrow area and form a strengthening structure.

Bone is continuously being broken down and remodelled by dedicated cells called osteoclasts and osteoblasts. Trabecular regions of bone have a far greater surface area for a given BMD when compared to that of cortical bone, leading to a more active remodelling metabolism. As such, the internal trabecular regions of bones like the vertebral bodies or the femur are far more likely to show a significant manifestation of osteoporosis, becoming less dense and more porous. The measurement of bone mineral density with a view to detecting the onset of osteoporosis might therefore be improved if the density of trabecular bone were measured in isolation.

Measuring Bone Mineral Density

There are numerous techniques that have been used to measure the density of bone. The many various techniques are based on a diversity of different aspects of radiation physics. The key methods are noted briefly. A detailed introduction to BMD measurement may be found in a review article by Speller et al (1989).

A series of radiographs, usually of the hands, are taken over a period of time for the BMD measurement techniques of radiogrammetry and photodensitometry. For radiogrammetry, the thickness of the cortical bone is measured from each radiograph. Subsequent radiographs may show a decreasing thickness of cortical bone, indicating a loss in bone mass. With the photodensitometry method, optical densities are measured from the radiographs to determine mineral density, via a calibration phantom.

A measure of BMD may be obtained by employing photon absorptiometry methods, in which a collimated photon beam from an isotope source of radiation is directed through the site of interest. The subsequently measured photon attenuation is related to BMD. The site to be measured must be immersed in water to eliminate the effects of the soft tissue surrounding the bone. A variation of this method has led to the development of dual energy x-ray absorptiometry (DEXA). The isotope is replaced with an x-ray source, which is used to generate beams of two different energies. The use of two energies removes the necessity for water immersion.

Other BMD measurement techniques include neutron activation and ultrasonic attenuation. Neutron activation can be used to determine the quantity of a particular element within the site of interest by measuring the quantity of g-rays generated from n,g reactions. Alternatively, since ultrasound waves are attenuated when directed through tissue, the tissue composition may be determined by the degree of attenuation. Bone with a lower mineral density will attenuate the ultrasonic waves to a lesser degree than healthy bone, providing the necessary BMD information.

Data from quantitative computerised tomography (QCT) images (reconstructed from x-ray profiles of the site of interest) can be related directly to bone mineral. It is thought that this ability to isolate the QCT response of trabecular bone offers an improvement in accuracy over the previous techniques that measure total bone mass (trabecular and cortical bone). As previously noted, osteoporosis manifests itself more readily in trabecular bone, so removing the effect of cortical bone will increase the sensitivity of the detection.

A New Method for Bone Mineral Density Measurement

 Research is being conducted into a method that isolates BMD measurement to trabecular regions of bone only. It utilises the coherent scattering of x-ray photons and is known as energy dispersive x-ray diffraction (EDXRD).

The EDXRD technique allows a scatter angle to be defined. A solid scatter angle produces a scatter which may be accurately positioned to fall within a volume of interest. For the purposes of EDXRD BMD measurement for osteoporosis detection, the scatter volume is chosen to fall within a site of trabecular bone.

An EDXRD experimental system is shown schematically in Figure 1. A polyenergetic source of x-ray photons is generated using an x-ray tube. These photons are collimated into a fan beam with primary lead slit collimators, which is set to be incident on a sample under investigation. Within the sample, some coherent scattering will occur, an optimum of which will take place at a fairly shallow experimental scatter angle f. Collimators are arranged at this angle (from the incident beam) so that only photons scattered at the chosen angle f will reach the detector. It is this arrangement that defines the scattering volume. The final beam generated with this geometry is detected with a high purity germanium detector and then processed, using a multichannel analyser.

The optimum experimental scatter angle f will depend on the molecular structure of the sample material under investigation and the wavelength/energy of the incident photons. Since the bone mineral in the trabecular sample has a crystal-like structure, Bragg’s Law will describe the conditions under which the constructive interference of x-ray photons occurs (equation 1).

nl   =  2d sinn                    (Equation 1) 

l is the wavelength of the photons, d is the spacing between the scattering planes, q is the Bragg scatter angle (half the experimental scatter angle f) and n is the order of diffraction.

Constructive interference of the photons will occur when the wavelength is a whole multiple of
2d sin q.

Figure 2 shows when this condition is satisfied for a given Bragg scatter angle q and scatter plane spacing d. The paths of three photons are shown before scattering (x, y, z) and after scattering
(x’, y’, z’).

There may be many scattering planes within a given material, presenting a range of plane spacings d. Some wavelengths from the energy range of the incident spectrum will satisfy Bragg’s Law for constructive interference (for a particular value of d). This leads to the detection of a spectrum of photon energies that will have a unique ‘signature’ related to the material under investigation, in this case trabecular bone (predominantly formed from hydroxylapitite). This signature spectrum will change for a different material because the spacings d will change. Peaks within the signature spectrum will be at energies that correspond with materials present within the scattering volume. The intensity of each peak can be related to the quantity of each material.

An EDXRD spectrum for a bone phantom is shown in Figure 3. The phantom consists of a mixture of cleaned, ground bone and fat, which together simulate in vivo bone and marrow. The spectrum was produced using the experiment illustrated by Figure 1. In this example, the peak at 27.5 keV predominantly represents the scatter contribution from the fat content of the phantom. The small peak at 33 kVp and the large peak at 40 kVp predominantly represent the scatter contribution from the hydroxylapitite content of the bone.

It is the intensities of the peaks for bone material within the spectra that is thought may be analysed to provide an accurate measure of bone mineral density within the scatter volume. If the scatter volume is positioned within a region of trabecular bone, a value for trabecular BMD may be used to provide an accurate indication to the presence or severity of osteoporosis.

EDXRD research currently being implemented at the City University Radiation Laboratory is concentrating on finding the minimum detectable limits of the technique. This involves work with bone samples, taken from total hip replacement operations, to simulate very small amounts of uniform BMD loss and the design of an optimised EDXRD system which will be compared to currently used methods such as DEXA. An attempt will also be made to relate small BMD losses as well as trabecular architecture, to changes in trabecular bone strength.

It is hoped that the EDXRD method can be shown to be sufficiently accurate, and of sufficiently low radiation dose, to be ultimately accepted in the clinical environment for BMD detection. If it can be shown that EDXRD can be used to detect smaller BMD changes than methods currently in use, the onset of osteoporosis in a patient might be detected earlier. Thus a treatment programme could be started sooner, perhaps preventing debilitating fractures that are also expensive to treat..

References

Farquharson, M. J., Luggar, R. D. and Speller, R. D. 1997. Multivariate calibration for quantitative
        analysis of EDXRD spectra from a bone phantom, Appl. Radiat. Isot. 48, 1075-1082.

Farquharson, M. J. and Speller, R. D. 1998. Trabecular bone mineral density measurements using
        energy dispersive x-ray diffraction (EDXRD), Radiat. Phys. Chem. 51, 607-608.

Speller, R. D., Royle, G. J. and Horrocks, J. A. 1989. Instrumentation and techniques in bone
         density measurement, J. Phys.E: Sci. Instrum. 22, 202-214.