Calcified Tissue International

, Volume 72, Issue 6, pp 745–749

Volumes From Which Calcium and Phosphorus X-Rays Arise in Electron Probe Emission Microanalysis of Bone: Monte Carlo Simulation

Authors

    • Department of Prosthetic DentistryThe Eastman Dental Institute, University College London, Gray’s Inn Road, London WC1X 8LD
    • Department of AnatomyUniversity College London, Gower Street, London WC1E 6BT
  • A. Boyde
    • Department of AnatomyUniversity College London, Gower Street, London WC1E 6BT
Laboratory Investigations

DOI: 10.1007/s00223-002-2010-9

Cite this article as:
Howell, P. & Boyde, A. Calcif Tissue Int (2003) 72: 745. doi:10.1007/s00223-002-2010-9
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Abstract

Monte Carlo simulations of trajectories for electrons with initial energy of 10 keV through 30 keV were used to map the 3D location of characteristic x-ray photon production for the elements C, P, and Ca until the electrons either escaped as backscattered electrons (BSE) or had insignificant energy. The x-ray production volumes for phosphorus slightly exceed those for calcium, but both greatly exceed the volume through which BSE travel prior to leaving the sample. The x-ray volumes are roughly hemispherical in shape, and the oblate spheroid from which BSE derive occupies only the upper third to half the volume of x-ray generation. Energy-dispersive x-ray emission microanalysis (EDX) may not be secure as a method for the quantitation of BSE images of bone in the scanning electron microscope (SEM). Ca:P elemental ratios from EDX analyses may also be imperfect.

Keywords

Ca:P ratioEDXBSE SEMBoneExcitation volumeEmission volumeMicroanalysis

Wavelength- and energy-dispersive electron probe x-ray emission microanalysis (WDX, EDX) have been utilized in large numbers of studies of Ca and P in the mineralized skeletal and dental tissues over four decades. Frequently, an attempt has been made to determine Ca:P ratios. The implicit assumption is then made that the excitation volume for PK (2 keV) and CaKα (3.7 keV) photons are identical. However, it would be expected to be the case that the cut-off limits, beyond which no further x-rays are generated, should to some extent differ since PK is excited by lower energy electrons which exist in the periphery of an electron diffusion volume of greater depth and radius. This problem is complicated by those of real structural dimensions in analytical bone samples, where, for example, exciting electrons may exit or enter tine ‘trabeculae’ (thin rods or sheets of bone tissue) at rather large distances from the beam impact point. How large such distances are will also depend on the quality of any embedding. If not embedded and if marrow spaces are present, then exit distances may be hundreds of microns, but we shall not consider unembedded or porous samples here. Because of the large diffusion volume for electrons having sufficient energy to excite x-ray photons, serious microanalytical work is conducted with sections thinner, or particles smaller than the excitation volume. Nevertheless, there are still numerous reports that use bulk samples such as those used in quantitative backscattered electron (qBSE) image analysis.

qBSE does not provide elemental analysis, but numbers of BSE exceed those of x-ray photons by several orders of magnitude, and in studying how much mineral is present per unit volume, is the better method. However, there are no absolute ways of calibrating either x-ray or BSE production in terms of mineral (or calcium or phosphorus) content. EDX and WDX spectroscopic analyses are conducted with reference to standards of known composition. qBSE employs known standards, but these do not contain the same elements as bone [1, 2].

Some workers in the bone field have proposed that qBSE be calibrated via the EDX signal for calcium [3]. We examine whether this is sensible, bearing in mind the poor statistics for x-ray counting compared with BSE detection. Furthermore, we calculate the absolute and relative volumes from which BSE, PK, and CaKα x-rays originate in a continuous, uniform, and massive bone sample, again noting that problems would be worse with real-world samples. To anticipate, we find that these volumes are significantly different, and conclude that this should be borne in mind when evaluating EDX data from bone or similar materials. Much of the relevant theoretical and practical literature is to be found in specialist microscopy and microanalysis journals. We are not aware of any prior study of the type undertaken here and hope it will be of general interest to those in the calcified tissues research field.

When an electron of known incident energy enters a solid sample it undergoes a series of scattering events with the constituent atoms. Some of these events are elastic, resulting in no energy loss during the interaction, but electrons are considered to lose energy between interactions following the Bethe approximation [4]. Infrequently, the scattering events are inelastic and arise in substantial energies being given up to the interacting atom, which then results in an x-ray photon with an energy characteristic of that element. Whereas these events are well understood for single elements and for simple binary and ternary compounds, usually metals or ceramics, they are less well understood for complex, multi-element, low mean atomic number mixtures found in biological systems.

Method

Understanding the quantitation of backscattered SEM images of bone has been assisted by the use of Monte Carlo modeling of electron-solid interactions [5, 6]. The Monte Carlo model used in this paper is based on the single selected atom approach published earlier, with the addition of inelastic scattering into the model. Elements are considered as transparent to their own characteristic radiation, and we ignore the minuscule fluorescence absorption of Ca x-rays to excite PK photons. We treat all x-rays as collectable, whereas in reality the solid angle presented by the x-ray detector is relatively small and only a small solid angle can be collected. In comparing BSE and x-ray data, however, we have taken into account the collection geometry.

When an electron enters a substrate it starts with an initial energy and then slowly loses energy by being scattered on its passage through the material. This energy loss is usually considered not to occur during the numerous interactions with individual atoms, where the collision is said to be elastic, but as a continuous process between scattering events [4]. Less commonly, inelastic scattering events may occur and a substantial portion of the incident electron energy may be lost during this single event. The interacting atom absorbs the energy which it can then release, amongst other processes, as an x-ray quantum characteristic of that element. Finally, the electron will either have lost sufficient energy to be captured by the sample or have returned to and escaped from the surface as a backscattered electron. The probability of either an elastic or inelastic scattering event taking place depends upon the relative scattering cross-sections that the atom presents to the electron [5, 7, 8, 9].

The Rutherford elastic cross-section (σR, in cm2/atom) may be determined by the formula of [10]:
https://static-content.springer.com/image/art%3A10.1007%2Fs00223-002-2010-9/MediaObjects/eq1.gif
where E is the energy of the electron in keV, Z is the atomic number, and α is the screening factor after Bishop [11] and Henoc and Maurice [12], and the Mott elastic cross-section by the introduction into equation (1) of two additional parameters (Gauvin & Drouin [13]).
For an electron energy, E, greater than ECrit, (the energy of the characteristic Kα x-ray quanta for that element), where U = ECrit/E, the inelastic cross-section is calculated after Green and Cosslett [14] by:
https://static-content.springer.com/image/art%3A10.1007%2Fs00223-002-2010-9/MediaObjects/eq2.gif
The total elastic or inelastic scattering cross-section for a material, σ, is calculated as the sum of the atomic fractions, ai, of the constituent element’s elastic cross-sections, σi:
https://static-content.springer.com/image/art%3A10.1007%2Fs00223-002-2010-9/MediaObjects/eq3.gif
The total scattering cross-section for the material or compound as a whole is then the sum of the elastic and inelastic cross-sections:
https://static-content.springer.com/image/art%3A10.1007%2Fs00223-002-2010-9/MediaObjects/eq4.gif
The selection as to whether a scattering even is elastic or an inelastic is determined from (3) by a random number, R1, generated using a routine by Press et al. [15].
https://static-content.springer.com/image/art%3A10.1007%2Fs00223-002-2010-9/MediaObjects/eq5.gif

Once the nature of scattering interaction has been determined, the specific element to be involved in the scattering process is selected via a second random number and in proportion to the relative contribution of each element in the compound or mixture to the relevant total scattering cross-section.

The electron enters the bulk substrate with a selected initial energy of 10, 15, 20, 25, and 30 keV. Each electron trajectory is followed, scatter by scatter, until either the electron returns to the sample surface and exits as a BSE, or it is considered to have had been captured by the sample when its energy has fallen below 250 eV (the excitation energy for Carbon K radiation is 282 eV): the latter are designated non-backscattered electrons (NBS) in Table 1 and Figure 1. The bone substrate was modeled with low, medium, and high levels of mineralization [5]. For each of the 15 scenarios of composition and initial electron energy, electron trajectories were followed until 105 CaKα x-ray quanta had been produced. The XYZ location of each characteristic x-ray quantum was recorded and used to determine the radius and depth ranges and total x-ray production volume for each element present. The volume traversed by BSE and NBS was determined by noting the location of the individual scattering events for a subset of 5,000 NBS and 1,000 BSE trajectories.

Table 1

Radius and depth of volume traversed by BSE (and NBS electrons) in 40% by volume mineralized bone for five accelerating voltages

https://static-content.springer.com/image/art%3A10.1007%2Fs00223-002-2010-9/MediaObjects/table1.gif
https://static-content.springer.com/image/art%3A10.1007%2Fs00223-002-2010-9/MediaObjects/fig1.jpg
Figure 1

Volumes from which phosphorus and calcium x-rays and BSE arise in bone mineralized to 40% by volume at (a) 20 kV and (b) 30 kV accelerating voltages. Absorbed, non-backscattered electrons designated NBS.

Results and Discussion

The present model, which included inelastic scattering, gave values for the BSE coefficient similar to those found in our previous work which utilized elastic scattering alone. The number of BSE varies with accelerating voltage and degree of mineralization of the bone, and the data confirm our current understanding and interpretation of BSE images.

The BSE traverse a layer close to the surface of the substrate whose depth is approximately 35–40% of that reached by electrons which may excite x-rays (Fig. 1a). The volume explored by BSE increases markedly with an increase in the initial energy of the incident electron (Fig. 1a,b), but is relatively constant for any one accelerating voltage for each simulated degree of bone mineralization (Table 1).

The number of incident electrons needed to produce a single x-ray quantum varies with the selected element, the accelerating voltage, and the degree of bone mineralization (Table 2). The Ca:P x-ray ratio varies with accelerating voltage, but little with the level of mineralization (Table 3).

Table 2

The number of phosphorus K and calcium Kα x-rays produced per 100,000 incident electrons for bone mineralized to 30, 40, and 50% by volume for five accelerating voltages

https://static-content.springer.com/image/art%3A10.1007%2Fs00223-002-2010-9/MediaObjects/table2.gif
Table 3

The ratio of CaKα:PK x-ray quanta for three levels of mineralization and for five accelerating voltages

https://static-content.springer.com/image/art%3A10.1007%2Fs00223-002-2010-9/MediaObjects/table3.gif

The phosphorus x-ray production volume was consistently greater than that for calcium (Table 4). The relative difference in these volumes decreases with increasing accelerating voltage, but varies only slightly with different levels of bone mineralization. The number of x-rays produced is not uniform over the whole of the x-ray production volume shown in Figure 1: in section, the intensity diagram is butterfly-shaped with wings directed downwards and laterally away from the site of the incident electron beam (Fig. 2).

Table 4

The dimensions of the x-ray production volume for bone (mineralization level 40% by volume) for phosphorus K and calcium Kα x-ray quanta

https://static-content.springer.com/image/art%3A10.1007%2Fs00223-002-2010-9/MediaObjects/table4.gif
https://static-content.springer.com/image/art%3A10.1007%2Fs00223-002-2010-9/MediaObjects/fig2.jpg
Figure 2

Volumes from which 33%, 50%, 66%, 95%, and 100% of the yield of calcium Kα x-rays derive in bone mineralized to 40% by volume at 30 kV accelerating voltage.

The overall efficiency for the collection of x-ray photons or BSEs is dependent upon the yield per incident electron, the proportion of that yield which can enter the detection system and the efficiency of the detector system in producing a measurable output. Taking into account the detector geometry and efficiency, the relative efficiency of BSE over x-ray detection for calcium is 4,354 fold at 10 kV, 8,544 at 20 kV, and 37,763 at 30 kV accelerating voltage.

The topographic relief formed on a bone sample as a result of polishing is such that lamellae within which the collagen approaches the surface more perpendicularly are ‘harder’, i.e., more resistant to polishing wear: conversely, collagen parallel with the surface is removed at a greater rate during polishing [16]. The resultant relief causes an increase in electron escape possibilities on ridges, and a reduction in valleys [6]. Since fewer electrons leave the surface in the latter, more are captured and available for x-ray production. Thus, it is certain there will be a local modulation in x-ray output which might be enough to show as an apparent artefact of more (valleys) and less (ridges) well mineralized lamellae, depending upon the quality of the sample surface finish. BSE imaging—which is radically more sensitive than x-ray emission imaging—fails to show apparent differences in lamellar mineralization if the sample surface is reduced to extreme flatness by micromilling: this is why we prefer micromilling as the basic preparative process for qBSE.

Now consider the influence of sub-surface features. In bone, these will mostly be osteocyte lacunae and Haversian canals, both of which should be filled with low density and low mean atomic number embedding resin. If the beam impacts first in a lacuna, it will be almost entirely trapped in the sample and all the electrons can be used in x-ray production, which will therefore be locally increased. Depending upon the x-ray take-off angle (the angle made between the line of sight to the x-ray detector and the sample surface), there may be additional complications due to self absorption of the x-rays, but this will generally not be important in bone-only samples (except that calcium atoms will significantly intercept phosphorus x-rays with grazing take-off geometry). It will, however, be extremely important near interfaces of bone with other materials, such as bone cements and implants.

Such border territory can be a conceptual minefield. In studying skeletal implants, we note that TiK radiation will excite Ca and P x-rays by fluorescence, and the sideways scattering of electrons by this metal will further cause a lateral extension of the x-ray production volume. If titanium is next to plastic, Ca and P x-rays may be excited at 50 µm or more from the beam impact point in a 50 kV accelerating voltage microanalysor. Conversely, impact on bone next to plastic will allow sideways-scattered electrons to hit Ti, which will be ‘found’ in the bone, where it is not.

Acknowledgements

This work was supported by INTAS grant 31864.

Copyright information

© Springer-Verlag 2003