Demonstration of the Differentiation
of Two Calcium Phosphates
by Soft X-Ray Microscopy
S. J. Bellamy1 , C. J. Buckley1 , X. Zhang2 , N. I. Khaleque1
1
2
Department of Physics, King’s College London, Strand,
London WC2R 2LS, UK
Department of Physics, SUNY at Stony Brook, Stony Brook,
NY11793, USA
Abstract. The feasibility of distinguishing between biological calcium
phosphates in calcified tissues by imaging thin sections with the soft
X-ray STXM is demonstrated using a sample containing synthetic phosphate powders. The strategy for quantitative processing of the images
acquired at selected X-ray energies is discussed and emphasis is placed on
diagnostic tests to detect thickness-effect distortions and other problems.
1
Introduction
Soft X-ray microscopy is a technique uniquely suited to the examination of calcified tissue. The CaL absorption edge falls between the CK and OK edges so
that contrast between calcium containing compounds and an organic matrix is
readily obtained by appropriate selection of imaging energies. This technique
has been refined to produce quantitative, low noise mass-thickness maps [1].
The CaL absorption edge is dominated by near edge features in the form
of a series of sharp resonance peaks. The intensity and location of these has
been modelled by considering the effect on the Ca2+ ion of the electrostatic
‘crystal field’ produced by the neighbouring ions in the mineral lattice [2]. Differences between the mass absorption spectra of some biologically significant
calcium phosphates are found experimentally [3]. Techniques for the quantitative separation of organic species using the fine structure at the CK edge are well
established [4].
To demonstrate the separation of calcium phosphates, calcium hydroxyapatite and calcium pyrophosphate were chosen because the fine structure of
spectra show clear differences. Moreover, calcium hydroxyapatite is the major
mineral component of bone and calcium pyrophosphate dihydrate has been associated with a deposition disease within bone [5]. Apart from investigation of
disease, potential applications include the interaction of ceramic implants with
tissue at the cellular level [6].
2
Methods and Materials
Small amounts of the calcium phosphates were placed together in the tip of an
embedding capsule, which was filled with ‘LR White’ methacrylate resin and
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cured overnight at 60◦ C. After trimming, sections were cut from the block with
a diamond knife, floated onto a water trough, then transferred to the well in the
back of a silicon nitride window and allowed to dry.
Fig. 1. Energy location of images near CaL
The images were acquired using the Stony Brook Scanning Transmission XRay Microscope (STXM) [7], at the National Synchrotron Light Source (NSLS)
[8], with a step size of 0.1µm and a dwell time of 10ms. I0 values of 0.9 ×
103 to 2.6 × 103 photons per pixel were recorded. Images were acquired at the
CK and CaL pre- and post-edges for quantification and at five energy locations
near the CaL edge to provide the chemical contrast, as indicated in Fig. 1. The
cross-sections shown for reference are approximate, resulting from an attempt
to reconcile total electron yield and transmission spectra from the two minerals.
The energy locations are again estimates, based on an attempt to match crosssection estimates for both minerals simultaneously and differ systematically from
the nominal monochromator setting. For convenience of reference, the peaks are
divided into two groups. The lower energy group is labelled ‘A’ and the upper
‘B’. The lowest energy location in each group is referred to as the peak and the
next, the trough. These are labelled APk and ATr , respectively, for the A group.
Above BTr is a high calcium cross-section location, labelled B2 , giving rise to an
image used for noise reduction [1] and as a mass-thickness reference.
Differentiation of Two Calcium Phosphates
3
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Image Processing
After conversion of the raw images to registered optical density maps, there are
four main stages. Firstly an approximate segmentation, to produce a preliminary
separation of the minerals. Then using the resulting spatial masks, estimation
of the cross-sections for the minerals separately at the various energies. These
cross-sections are then used to fit mass-thickness values for the three components
(embedding resin and the two calcium phosphates) at each pixel location. The
final stage is diagnostic, involving the examination of residuals and detection of
distortions due to thickness effects.
14
12
10
0.6
Freq. Arb.
Corrected ATr OD
0.8
0.4
8
6
0.2
4
0.0
2
0.0
0.5
1.0
Corrected B2 OD
1.5
0
-0.1
-0.05
0.0
0.05
0.1
Mean Reduced Angle [ radians]
Fig. 2. Generation of mean reduced angle Map for ATr location.
Extensive use is made of scattergram techniques to visualise the relationship between sets of maps [9]. Figure 2 shows the pre-edge subtracted, carbon map corrected [1], optical density map for the ATr location plotted against the B2 mass
reference scale. Of the four locations (APk , ATr , BPk and BTr ) this scattergram
shows the most obvious separation of the hydroxyapatite and pyrophosphate
regions. The dense cluster around the origin corresponds mainly to noise associated with the clear embedding resin pixels. The latter can be excluded by means
of a simple threshold. Now, referring to Fig. 2, consider the polar angular position of a pixel about the origin. This is a function of the mineral cross-section,
with the mass thickness divided out. A histogram of the angular values, after
subtracting the mean, is shown in the right hand panel. The distributions for the
two minerals are poorly separated. There are four of these mean reduced angle
maps. To reduce this to a more manageable dimension, a principal components
analysis is applied to the variance-covariance matrix [10].
Over 90% of the variance was accounted for by the first three principal components. Of these, only PC2 contains contrast information, the other two are
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mostly noise. Simple numerical modelling confirms this interpretation, but indicates that the noise level is greater than would be expected from photon statistics, perhaps because of misregistration. The variable loadings for PC2 indicate
that the difference between the peak and trough cross-sections provides the required contrast.
To produce the approximate segmentation, a three-way scattergram is considered with co-ordinates consisting of the corrected B2 mass thickness reference,
together with the A and B group contrast optical densities. Projecting onto a
hemisphere with the the mass thickness reference as the polar axis (clear embedding medium pixels being excluded) gives a two-way scattergram with two
overlapping clusters. Choosing a line through the waist generates a pair of spatial masks, approximately separating the minerals. The mineral cross-sections
were then estimated by considering the set of two-way scattergrams between
the corrected optical density images and that of the mass-thickness reference,
with one of the approximate mineral selecting masks applied. Again considering
the angular position of each pixel, there is a median value for each scattergram.
Choosing a certain quantile about this median for each scattergram and taking
the intersection over all the scattergrams identifies a set of pixels, which may be
regarded as ‘typical’ in the sense that they remain near the median, irrespective
of disturbance by noise. A proportional line is fitted through these pixels giving a
gradient and error estimate. Since the CaL cross-edge map is included in the set
of scattergrams, the mineral cross-sections at each energy can be deduced from
tabulated values for calcium at the pre-and post-edge locations. This procedure
is robust, in that it is insensitive to the tails of the optical density distributions.
However the choice of the size of the quantile needs further investigation.
The three mass thickness maps are generated from the seven pre-edge subtracted optical density images using a pseudo inverse from the singular value
decomposition [11], the most reliable method for an overdetermined system. In
fact the estimated relative uncertainty in the cross-sections was found to be comparable with the relative size of the singular values, so that the signal-to-noise
ratio is barely satisfactory.
The mass thickness maps are examined for significant regions of negative
values and of overlap of the mineral phases, Fig. 3. Comparison with the visual
appearance of the field and the preliminary segmentation are made. The optical
density residue maps are then calculated. Inspection readily reveals misregistration in the original images. Plots of residues against estimated optical densities
are examined for trends and offsets from zero which may be interpreted as errors
in the cross-section estimates. Detection of thickness effect distortions depends
on identifying curvature of two-way scattergrams in the presence of broadening by noise. The curvature will be such that the high cross-section reference
optical density becomes less than the trend line at higher values. In projected
three way scattergrams, clusters become extended in the co-latitude direction.
These effects were found with an additional image located at higher energy than
B2 , with greater Ca cross-section. Further, it is helpful to relate pixels in these
suspect areas to their location in the specimen, which contains creases and folds.
Differentiation of Two Calcium Phosphates
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15
m
10
5
0
-0.01
0.0
0.01
0.02
0.03
HAP Thickness [ m ]
-0.02
0.0
0.02
0.04
0.06
Pyro Thickness [ m ]
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
Embed Thickness [ m ]
Fig. 3. Thickness maps from mass thicknesses using bulk densities.
4
Conclusions
The method briefly outlined here has proved effective in separating two calcium
phosphates. The synthetic specimen has the unexpected feature that the mineral areas are overlaid with resin, the ratio of mineral to organic matrix mass
thicknesses being similar to natural bone. The method differs from that of Zhang
[4], in that the latter requires cross-sections at contrast locations from spectra.
However, that method has proved successful in treating mixtures of organic compounds, whereas the present method has only been applied to separate minerals.
Further work is required to determine the performance in the presence of
mixtures, perhaps by numerical modelling. The diagnostic methods need much
development as does the estimation of cross-sections. A further consideration is
whether or not to apply pre-edge subtraction to the optical density images for
the fit.
Acknowledgements
The authors would like to thank Steve Hulbert and his colleagues in Beamline
Development, NSLS, for assistance with the spectroscopy which underpins this
work and Sandra Downes, (RNOH) for the phosphate samples. SJB would like
to thank the EPSRC for funding the research trips to the NSLS.
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