1. Introduction

X-ray projection imaging has been an invaluable technique for the study of mineralized biological tissues, specifically bones and teeth, for more than a century [1]. With multiple projections from at least 180 degrees, tomographic absorption three dimensional images are routinely created (usually by the filtered back-projection method [2]). Tomographic techniques have become the mainstay for visualizing and interpreting the structures of mineralized tissues over the last two decades [3, 4], and the quality of both signal and resolution is ever increasing [5, 6]. Phase sensitive imaging techniques, such as diffraction enhanced imaging [7] or x-ray interferometry [8] have expanded the tomographic possibilities by enhancing interfaces and different material-phases, known to exist within many biological tissues [9, 10]. Yet phase-enhanced imaging has gone much further than simply providing qualitative results. Propagation based images have recently been used to numerically retrieve the phase-shift of hard x-rays transmitted through imaged samples [11, 12]. As a result, refined details of structural variations at the micron length-scale are at hand, mainly due to Fresnel propagation over moderate sample to detector distances. Fresnel fringes are nevertheless considered to be unwanted artefacts because they distort the image, making it difficult to obtain the correct attenuation coefficients.

The need to use high-fluxes of x-ray sources that are partially coherent and the technically stringent arrangements of the x-ray optical setups appear to limit the use of phase-enhanced imaging for the study of complex and hierarchical biogenic materials [13]. However, the full field quasi non-destructive virtues of these techniques show great promise for furthering our understanding of the materials found in teeth and bone. The weak interactions of x-rays with all biomaterials is the main advantage of using x-rays to probe and investigate mineralized tissues. At energies commonly used for tomography (E>8 keV, i.e. λ<1.5 Å), the refractive index (n) resulting from absorption and scattering of incoming photons is almost unity, and may be written as:

where the imaginary component β is due to the photoelectric effect and inelastic Compton scattering and the real decrement δ is due to Thomson scattering [14]. For the combination of elements constituting human hard tissues, namely calcium phosphate salts (carbonated apatites), β is 2–3 orders of magnitude smaller (O≈10^-9) than δ. For conventional tomography, β is readily obtained from simple absorption images and it is usually evaluated in the form of the linear attenuation coefficient. However for imaging of structural details in materials and tissues of similar composition, retrieving δ is considerably more informative. The phase shift effect on the propagating wave is far greater than the absorption effect and this can be exploited if the phase shift can be determined.

For the retrieval of δ it is necessary to determine phase shift differences across the propagating x-ray field. Using interferometric methods, such differences can be determined if at least partially coherent illumination is used, as is available in 3^rd generation synchrotron facilities [15] and certain lab-sources [16, 17]. Three experimental approaches have been reported: X-ray interferometry, diffraction enhanced imaging and Fresnel-propagation based phase imaging (e.g. holotomography). For overviews, see [8], [18], [19]. It is possible to determine δ with all these approaches, each having significant benefits, but Fresnel-propagation imaging clearly has the advantage of employing the simplest component setup: phase modulation is achieved via Fresnel propagation by changing the sample to detector distance.

In this work, we present an analysis of the propagation of x-rays through a moderately complex biomaterial: tooth dentin. Dentin is considered to be a highly organized variant of the bone family of materials [20] forming the major bulk of all human teeth. This material is characterized by having tubules extending more than 1 mm in length, running more or less parallel to each other across the entire dentin thickness. In tooth crowns most tubules have sheaths of peritubular dentin (PTD) which is dense and highly mineralised, forming cylinders spanning 2–5 microns and housing empty voids of 1–2 microns in diameter (in life these cores are filled with living soft tissue). The tubules are embedded in a less highly mineralised intertubular dentin.

To interpret the interference patterns produced by a partially coherent wave propagating through the quasi-ordered tubular structures of dentin, we derive a simple analytical solution which we further interpret by simulations and analysis of experimentally recorded tomograms. Our results show how structural insights can be obtained by interpreting the outcomes of propagation for the purpose of performing 3D virtual microscopy in teeth.

2. Fresnel imaging using hard x-rays

A complete development of the theory and application of Fresnel-propagated tomographic x-ray imaging is beyond the scope of this paper (for reviews see [21–26]). However, some of the key concepts necessary for the analysis and interpretation of the data are presented here briefly.

It is convenient to maintain a spherical wave approach with a unit amplitude incoming x-ray wave (subscript inc) impinging on the sample:

The propagation axis from the source to detector is in the z direction. The image plane is described by the (x,y) coordinate system perpendicular to z. We limit the discussion to the spatially varying part of the wave, dropping any time dependant terms e^iωt , because we assume a monochromatic beam. λ is the x-ray photon wavelength and l is the source-to-sample distance (see Fig. 1). $\sqrt{x^2+y^2}\ll l$, the incoming spherical wave may be approximated to be a parabolic wave transmitted through the sample. Interferences occur within the transverse coherence length, which for our partially coherent source is:

with s_α the angular divergence of the source related to its lateral size s via s_α =s/l.

Upon transmission through a thin object sized several millimetres at most, the effect on phase and amplitude follows the thin lens approximation [27], and the x-ray field immediately downstream of the sample (subscript 0) can be described by:

Thus the effect of the sample on the wave is given by the complex transmission function T(x,y), which varies with the angle of projection when tomographic datasets are recorded. We note that ϕ and B represent the outcomes of propagation through the sample. They are related to the real and the imaginary parts of the refractive index n, as given by Eq. (1), obtained by line integrals of δ and β along the propagation direction z:

with ϕ ₀ the phase shift in vacuum (n=1) over the sample length.

The recorded intensity immediately behind the sample is the squared modulus of u₀ in Eq. (4), and appears as an absorption radiograph in which each point corresponds to I₀ (x, y)=e ^{-2B(x, y)}. Note that by convention in radiography, the negative terms of the exponents are used to create the images, rather than the actual squared exponentials. At moderate distances d beyond the sample (downstream), the Fresnel approximations hold, and the wave-front at a distance z=d takes the form of a Fresnel integral:

Thus, u_d (x,y) is essentially the Fresnel-propagated transmission function of the object, multiplied by a parabolic phase term (see Eq. (2)). Eq. (7) can therefore be written as the convolution of the transmission function T(x,y) with a complex propagation function P_D (x,y) in transformed coordinates. We introduce magnification M=(l+d)/l, d→D=d/M and obtain in the plane (X,Y)=(x/M,y/M):

In reciprocal space Eq. (8) assumes the familiar form of a multiplication (Fourier transforms are indicated by ~).

with fX and fY the spatial frequencies corresponding to X and Y in real space.

We proceed to use these equations to analytically interpret the propagation through samples with tubular inclusions (Fig. 1). We consider propagation parallel to a cross-section, which is perpendicular to the tubule long axis. For simplicity we align this axis with the y-axis of our coordinate system. As the circular symmetry holds along the entire y-axis, it is sufficient to consider a x-z slice across the tubule, i.e. y=0. Additionally, following a weak phase approximation [28], we assume e^x ≈1+x and proceed to linearize the transmission function from Eq. (4), thus T(x, y=0)=T(x)≈1+iϕ(x). With β≪δ, the tubules are thus assumed to be pure phase objects. Now, instead of developing the one-dimensional Fourier transform of ϕ(x) we calculate the transform of the map of its components δ(x, z) across the tubule, parallel to the x-z plane. The analytical transform of such a 2D circularly symmetric tubule cross-section is easier to calculate than the transform of its projection. An intuitive justification for taking this approach arises when considering the construction of a tomogram to be a process of image formation. Such a tomographic image, although virtual, may be imaged onto a detector array at d. Points on any plane in this 3D image can therefore be Fresnel-propagated along z.

 

Fig. 1. Schematic representation of x-ray propagation through a material containing dense tubular structures. A full tomographic dataset is obtained from 900 projection images of the sample, viewed from 180 degrees. The large source-to-sample distance coupled with a small beam divergence allow imaging of interference patterns of discontinuities due to propagation. Tomograms are produced by placing the detector in the standard absorption regime (d₀) and at multiple propagation distances ranging from the near Fresnel to the far Fresnel regimes (d₁–d₃).

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A formal argument arises in the Fourier projection-slice theorem [29, 30]. Simply stated, a one-dimensional Fourier transform of a parallel projection of a two-dimensional object is equal to a slice through the two-dimensional Fourier transform of the same object, orthogonal to the direction of projection. The tubular geometry of our samples facilitates treating all radial projections as identical both in real space and in the Fourier domain. This can now be further analyzed by treating tomography and propagation as two invariant linear transformations, the order of which we merely exchange [27]. For the simple case of projecting a cylinder with an empty core, consider the true 3D form of n(x, y, z) which (far from any absorption edges) can be seen as a 3D representation of the microstructural electron density (cf. Fig. 2(a)). The tubule is approximated to be a circular void with a dense collar of PTD, which is embedded in intertubular dentin matrix. Constructing the distribution of the object refractive index (Fig. 2(b)) in the x-z plane by superposing two circ functions with $r=\sqrt{x^2+z^2}$, where circ(r)=1 for all r≤1 and 0 for r>1, we write

where r_void <r_tubule are the void and tubule radii respectively, and n_α is the refractive index of the material α: n_α=1-δ_α+iβ_α with α={void,PTD, matrix}. From the map of the refractive index n(r) we can calculate ñ(ρ) by applying the Hankel transform of zero order:

where ρ is the conjugate variable to r in reciprocal space, we obtain:

with $\rho =\sqrt{f_x^2+f_z^2}$, J ₁ represents a Bessel function of the first kind order one and δ _Dirac is the Dirac distribution at ρ=0. The projection-slice theorem now states that ñ(ρ)=Ñ(f_x ) with Ñ(f_x ) the one-dimensional Fourier transform of the refractive index N(x)=∫n(x, y=0, z)dz projected along the z-direction (note that due to the circular symmetry the projection angle 0° is equivalent to all other angles). We now apply Fresnel propagation to Eq. (12) by multiplication with the propagator P̃_D(f_x, f_z )=P̃_D($f_x^{\mathit{2}}$${\mathit{+}f}_z^{\mathit{2}}$ )=P̃_D(ρ), as given by Eq. (9) and inverse transforming. While in Fourier space the result is a sum of two concentric oscillating damped Bessel functions of differing complex amplitudes, in real space the squared modulus of the inverse transform can result in a positive, negative or almost zero intensity distribution, as shown by Fig. 2(c). Shown are plots of a tubule with no PTD, tubule built only of PTD and the damped amplitude resulting from the cancelling-out of both contributions (propagation corresponding to d=433 mm).

 

Fig. 2. Model of the distribution of the refractive index n(r) in a single tubule and resulting fresnel propagated images. (a) Schematic of tubule (b) Schematic illustration of the real component of the refractive indices of void, PTD and matrix (c) The outcomes of propagation of a tubule with only a void (constructive interference), only PTD without void (destructive interference) and the sum of both. When the interference patterns overlap, image contrast vanishes. The analytical model (Eq. 12) was used to construct both contributions in Fourier space using Bessel functions. Propagation over d=433 mm was assumed, and the tubule was set to have a radius of r_tubule =1.67×r_void with r_void =0.8 µm. Pictures show the squared modulus of the inverse Fourier transform of the Bessel functions.

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To substantiate our result, we performed both numerical simulations and measurements of propagation. As shown in the next section, the results closely match the analytical solution and can therefore be used to interpret experimental results of human tooth dentin measurements. We therefore present a comparison between the outcomes of simulation and experimental results. Such a comparison is essential if Fresnel-propagated microtomograms are to be used for dentin microstructural investigations.

2. Experimental setup and data collection

Dentin samples were prepared from teeth that were extracted and discarded during routine dental treatment. Samples were prepared by cutting small parallelepiped cuboids that included both enamel and dentin (2×2×3 mm ³) spanning the so-called dentin-enamel junction (DEJ) as shown in Fig. 3(a) [31]. About 0.5 mm enamel and the adjacent 1.5 mm dentin beneath it were imaged (Fig. 3(b)). All samples were kept wet during preparation and during all stages of data collection, having been placed in small thin-walled cylindrical Plexiglas vials filled with water.

 

Fig. 3. Sample of human crown dentin with attached enamel: (a) Schematic illustration of the position and form of typical samples cut out of the intact tooth. (b) An experimental contact (absorption) radiogram of a sample prepared as shown in (a). The denser enamel appears brighter due to the convention of representing absorption images using a (-log) scale.

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2.1 Cross-sectional electron microscopy imaging

An additional sample was used to prepare several 2D slices across the dentin microstructure for scanning electron microscopy (SEM). The sample was dehydrated in a series of ethanol solutions, embedded in Polymethylmethacrylate, then sliced and polished orthogonal to the tubular orientation. This revealed the cross-section at a depth of approximately 500 µm below the enamel and DEJ (shown in Fig. 4(a)). The uncoated backscattered SEM image reveals a distribution of tubule sizes, with corresponding voids and mineral content.

2.2. X-ray data collection

For imaging dentin, we used the tomographic setup on the BAMline of the Berlin electron storage ring company synchrotron facility (BESSY [32]). Samples were placed on a rotation stage, located 35 m downstream of a 7-tesla wavelength shifter. With an approximate source full-width-at-half-maximum (FWHM) of 110 µm width and 70 µm height, the resulting lateral coherence length at λ=0.5 Å equals L_t ≈25 µm in the vertical direction and 15 µm in the horizontal direction. A double multilayer monochromator provides a high flux quasi-monochromatic beam with an energy distribution: ΔE/E≈10^-2 which has no effect on L_t . The 2D x-ray detector is mounted onto an air floating rack, aligned along the beam axis. This allows precision z displacements extending up to 1.12 m away from the sample. The beam is transmitted by the sample and propagates towards the detector where it impinges on a 22 µm thick CdWO₄ single crystal scintillator supported by a 0.3 mm thick Y₃Al₅O₁₂ (YAG) substrate. The parasitic light emitted by the YAG is filtered by a Schott glass-filter (cut-off at λ=4950 Å). A Rodenstock TV Heligon 1:1,0 (f=21 mm) objective and a Nikkor telephoto (f=180 mm) lens image the scintillated light onto a back illuminated Princeton VersArray 2048B CCD detector array with a pixel size of (13.5 µm)². We thus obtain magnified pixels detected from an area of (1.59 µm)² on the scintillator screen. The lateral optical resolution of the complete optical system is approximately 3 µm.

 

Fig. 4. Tubules in dentin as seen in scanning electron microscopy. (a) An image of 244 tubules seen in one of the backscattered SEM images of a polished slice of dentin. The inset shows a magnified tubule which includes a void and a dense peritubular sheath. (b) Distribution of sizes of void radii and (c) tubule radii. (d) Histogram of ratios of tubule and void radii. <r_void >=0.8 µm, <r_tubule >=1.9 µm. The highest frequency of r_tubule/r_void is a ratio of 2.4.

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2.3. Fresnel-propagated micro tomography experiments

The creation of one complete 3D tomogram requires obtaining parallel radiographic projections from 180° around the long axis of the sample. Projections were obtained every 0.2° in addition to multiple flat-field images that were acquired periodically by moving the sample out of the beam. The flat-field images are used to correct for background fluctuations of the inhomogeneous x-ray illumination. Radiographs were acquired using a 25 keV photon energy for samples placed at d₀=5 mm (absorption regime), d₁ =144 mm (“near” Fresnel regime) d₂ =433 mm and d₃ =720 mm (“far” Fresnel regime, see Fig. 1). These sample-to-detector distances were chosen to allow imaging of the full range of spatial frequencies in the structure [33]. The 900 angular projections were used for reconstruction by means of the filtered back-projection technique [2] using the open-source PyHST software (A. Mirone, SciSoft group ESRF, Grenoble, France). Fresnel-propagated tomograms were reconstructed without using the conventional negative logarithm scale normally employed for absorption tomogram representation. As a result, constructive interferences appear as bright patterns, and destructive interferences appear as dark patterns. A quantitative structural interpretation of the tomographic cross-sections thus requires a detailed analysis of the interactions of the beam with the underlying dentinal microstructures. We perform this analysis by simulating the propagation of a wave transmitted by a single tubule.

3. Propagation simulation and analysis

Prior to simulating the propagation through a virtual tubule, the inherent blurring of CCD camera, optics and scintillator screen need to be considered. These can be described by an overall point spread function assumed to have Gaussian shape with a FWHM of 3.0 µm in our setup. The partial coherence of the x-ray beam is described by the lateral extension of the source s and results in an additional blurring of the Fresnel-propagated images with an additional Gaussian kernel (sized s′ = s ’ d/l=110µm·0.433m 35m≈1.4µm at d=433 mm propagation distance). SEM images (Fig. 4(a)) were used to estimate the average radius of the voids <r_void > and the average radius of the entire tubule <r_tubule > (=PTD+void). The distribution of radii of voids and tubuli as well as their mutual ratios r_tubule/r_void are shown in Figs. 4(b), (c) and (d). Some 244 tubules were measured by binarization from which <r_void >=0.8 µm, <r_tubule >=1.9 µm were determined. These numbers are well within the range reported in the literature [34, 35]. The average distance between tubule centers was found to be 9 µm but this is known to vary [34]. From the distribution of ratios r_tubule/r_void , the ratio of 2.4 was found with the highest frequency, however a wide range of ratios is seen.

The values of δ and β that were used in the simulations were calculated for carbonated hydroxyapatite (Ca₁₀(PO₄)_5.5(CO₃)_0.5(OH)_1.9(F,Cl)_0.1 [36]) using from the public-domain ESRF XOP tool (XOP v 1.8, Shanze, ESRF, Grenoble, France). The density of constituents severely affects the calculated δ and β. For simplicity, all non mineralized points were assumed to be composed of water. For the PTD we used a density estimate of 2.6 gr/cc apatite, whereas for the interdental matrix we used 2.0 gr/cc [37]. We note however that a wide range of values has been reported in the literature [36–38]. We observed the PTD to appear remarkably similar to enamel when observed in backscattered SEM images at depths of 300–500 µm beneath DEJ (Backscattering SEM image gray values are well known to correlate with mineral density in apatite tissues [39]). We therefore used a moderate estimate, well below the ~3.0 gr/cc known for enamel. The parameters that were used are listed in Table 1.

Table 1. Values for the density and refractive index used for the numerical simulations

View Table

We simulated propagation of x-rays of a wavelength of 0.5 Å (25 keV) and the calculations were performed by numerically solving Eqs. (5,6,7) and calculating the resulting image intensity. A line across the virtual wave propagating through and beyond a tubule was plotted for a range of tubule to void radii ratios and also at various propagation distances. Figure 5 shows line profiles of propagation over a distance of d=433 mm for a void radius of r_void =0.8 µm and for a range of tubule radii. The results obtained for tubular radii ranging from 0.8 µm to 4.0 µm are presented in the form of a pseudo-3D surface plot. Note the resemblance to the analytical results of propagation of a x-z plane based on Eq. (12), shown in Fig. 2(c). When there is no PTD (r_tubule=r_void =0.8 µm) the void appears to focus the x-rays, forming a central peak. This is emphasized in the line profile (a) redrawn in the inset on the left hand side of the figure. As the thickness of PTD increases, focusing is reduced and at some critical thickness, the interference patterns vanish and no contrast is visible. The line profile is just about flat (Fig. 5(b)). This occurs at r_tubule =1.67×r_void which appears to be a critical ratio of void to tubule thickness, when 25 keV x-rays are used. Tubules of this diameter and thickness do not show detail in the reconstructed Fresnel-propagated tomograms due to a cancellation of the constructive and destructive interference contributions. By increasing the PTD thickness further (Fig. 5(c) calculated for r_tubule =2.5×r_void ) an inversion of the effects of constructive and destructive interference is seen. Thus for thick PTD, a central dark minimum surrounded by bright rings is observed.

Simulations of propagation through both smaller and larger void radii (well within the range observed from SEM pictures) revealed patterns that were not significantly different from those seen in Fig. 5. Thus tubules with none or little PTD always appear as brighter spots surrounded by dark bands while tubules with radii of 1.65~1.7 times the radius of the void show negligible contrast and tubules with even thicker PTD sheaths always appear as dark spots surrounded by bright circles.

 

Fig. 5. Simulation of the evolution of the interference pattern of a tubule as a function tubule/void radii ratios. The r_tubule/r_void ratio is calculated from 1 (corresponding to no PTD) to 5. Three characteristic line profiles are plotted for ratios of (a) 1, (b) 1.67 and (c) 2.5. With no PTD surrounding the void (a) we note a bright interference peak forms at the tubule centre while tubules with a very thick PTD (c) will cause a dark central interference. For a very critical ratio r_tubule/r_void ~1.67 we note that the interference pattern almost vanishes (b).

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The effect of increasing the propagation distance can be seen by the pseudo-3D projection shown in Fig. 6. When r_tubule=r_void , no PTD exists and the resulting interference patterns at distances up to 1.2 meters are shown in Fig. 6(a). In the absence of PTD the tubule functions as a focusing lens and increasing the propagation distance only highlights this effect. For tubules with thick PTD, i.e. r_tubule >~1.8×r_void , we observe a constant defocusing behaviour of the tubule, independent of the propagation distance. Fig. 6(b) shows the results for r_void =0.8 µm and r_tubule =2.4 µm. At propagation distances between 0 and 0.2 m we observe a strong increase in the signal amplitude and for propagation distances > 0.2 m the effect of propagation is basically to spread out the interference fringes (see line plots Fig. 6(b) 1–3). Interference patterns obtained for tubules with r_tubule ≈1.67×r_void (data not shown) show complex forms in the simulations. They appear as very small fluctuations of the amplitudes at increasing propagation distances with extremely low contrast and they are practically not detectable in experimental tomograms. A typical cross-section of a Fresnel-propagated tomogram is shown in Fig. 7(a). The distribution of structures is in stark contrast to the scanning electron images of the microstructure in a polished sample (Fig. 4). The primary difference arises from the effects of constructive and destructive interferences (both corresponding to voids in the tubules) that appear due to propagation. To fully interpret the results of simulation, we numerically propagate the SEM image shown in Fig. 4(a), to obtain a virtual Fresnel-propagated cross-section SEM image in Fig. 7(b). This was done by assigning complex refractive indices according to the grey values corresponding to tissue densities in dentin (PTD, water-filled void and intertubular matrix) as provided by Table 1.

 

Fig. 6. Simulated Fresnel propagation of tubules at various distances. The two graphs (a) and (b) show the effect the propagation distance has on the measured interferences. Graph (a) shows the evolution of contrast produced by a simple void of r_void =0.8 µm while in (b) a PTD sheath of 1.6 µm thickness was added (r_tubule/r_void=3). The three plots on the left show explicitly the interference patterns at (1) d=144 mm, (2) d=433 mm and (3) d=720 mm.

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Figure 7(b) was thus obtained by Fourier transforming a complex map derived from Fig. 4(a), multiplying by the propagator (Eq. (9)), inverse transforming and calculating the squared modulus. The detector resolution was taken into account by filtering the intensity image with a Gaussian kernel. Note the remarkable similarity to Fig. 7(a), which shows a slice through a reconstructed Fresnel propagated tomogram of dentin from the adjacent area in the tooth.

 

Fig. 7. Slice across a Fresnel propagated tomogram compared with a propagated SEM image. (a) A magnified region from a recorded Fresnel propagated micro tomogram approx. 0.5 mm below the DEJ (experimental data) shows striking similarity to the simulated picture (b) obtained by virtual Fresnel propagation of the SEM image from Fig. 4. Both pictures correspond to a propagation distance of d=433 mm.

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Areas of constructive and destructive interferences appear in some areas of the propagated SEM image, while in other areas contrast is lost.

Finally, the mean distance between the center of tubules must be kept in mind. This is important because imaging dentin beyond a certain propagation distance will cause the fringes to overlap, forming complex interferences. As a result structural detail will become obscured, as can be seen for example when comparing tomograms recorded at short and long propagation distances in Fig. 8. Notice how the same region in tomographic datasets obtained at different propagation distances (d=144 mm, 433 mm and 720 mm) reveals similar features, which appear to grow. The spreading of the interference patterns is seen to exceed the mean distance between the tubules, resulting in reduced visibility and blurring. The distance at which tubules can be visualized best is therefore a compromise between tubule size and intertubular distance, for different depths of dentin.

 

Fig. 8. The effect of increasing propagation distance on image quality. Identical regions from Fresnel-propagated micro tomograms of dentin showing a close packing of tubules. The images recorded at (a) d=144 mm and (b) d=433 mm reveal clear details of the microstructure. At d=720 mm (c) tubules lose detail in the image due to the overlapping interference patterns.

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4. Discussion

In this work we highlight the advantages of performing microtomography using Fresnel-propagated radiographic projections for investigating the tubular structures found in human dentin. Due to interference in quasi-coherent x-ray beams, the 3D distribution of tubules can be visualized in two forms: bright ‘spots’ surrounded by dark rings, when the PTD is absent or is thin (r_tubule <=1.5×r_void ) or dark ‘spots’ surrounded by bright rings when the PTD thickness is substantial (r_tubule >=2×r_void ). Tubules where r_tubule =~1.67×r_void show no features when coherent 25 keV x-rays are used for imaging. Thus the three-dimensional form of the tubules and deviations of the tubule design can be mapped within the bulk. Interference fringes originating from features only 1~2 micrometers in size, which is below the resolution of our system, are clearly observable in the Fresnel-propagated reconstructed tomograms. However this is only true as long as patterns from neighbouring structures do not overlap. At large (>0.5 m) distances, dentin features appear to touch and overlap (Fig. 8(c)) rendering such images less informative.

Our approach has been to consider the effects of thickness and structure of tubules in dentin on the tomographic image. This was done by comparison between an analytical model, numerical simulations of propagation and real measurements of tomograms of dentin microstructure. The enhancing effects of propagation, namely increased contrast of tubules and concomitant magnification of the lateral dimensions of features in the reconstructed tomograms, facilitate exploration of the subtle microstructure variation, as well as the distribution of defects. We thus believe that this approach has wide potential application.

Our results indicate that the tubules appear denser and thicker at increasing distances beneath the enamel. This is in notable accord with other reports of stiffening and hardening of the microstructure [31, 40, 41] which can be explained by both increased density and thickness of the PTD. From approximately 200 µm beneath the enamel cap and deeper in, clear interferences patterns are seen, many with dark centers. We note however, that clusters of dark spots surrounded by white rings are seen in some areas, whereas clustered white spots surrounded by dark rings are seen in other areas. Further analysis is still needed to fully characterize the distribution of the PTD and tubule thickness in teeth.

Under conditions similar to our experiments, namely sample sizes, detector resolution, coherence length and energy, it appears that propagation distances above 0.1 m and below 0.5 m are needed to obtain a good signal without obscuring structural detail. However significant regions in the volumes appear with little or no contrast, implying that many tubules are characterized by a ratio of tubule radius (r_tubule ) to void radius (r_void ) of approximately 1.67. The explanation for the importance of this critical ratio is provided by considering Eq. (12). When the amplitude contributions of PTD and void cancel each other (Fig. 2) an overall damped intensity arises. Thus n(r) essentially appears as n_matrix and the tubules are not visible at any propagation distance: propagation only modulates the phase of the transmitted wave. In order to reduce the ambiguity observed for such tubules, where the constructive and destructive interference patterns overlap, other tomographic datasets need to be obtained at higher energies. By changing the energy, we alter the x-ray interaction with the biomaterial, thus altering δ as well as the appearance and magnitudes of the Fresnel fringes.

We contend that Fresnel-propagated tomographic x-ray imaging offers a unique set of opportunities to probe and characterize complex biomaterials such as dentin. Clearly the well-known (a-priori) estimates of the sizes and densities of the different material phases were essential for the systematic analysis of the interference patterns in dentin. Thus, partial structural information was used to interpret the Fresnel-propagated images, which were then used for additional three-dimensional analysis and interpretation of the tubular dentin.

One other issue merits some additional consideration. Some uncertainty remains in assuming the mass densities of PTD and intertubular matrix. Backscattered SEM images suggest that in certain areas the PTD collar density is very similar to the density of the intertubular dentin matrix. Obviously the interference and intensity distributions thus become similar to those seen when the PTD thickness is reduced, as shown by Fig. 2 and 5. Thus more work is needed to determine the density of PTD and its variations through the structure.

Our approach does not go to the full extent of complete retrieval of the phase shifts from tomographic datasets from various distances (following [19, 33]). Multiple datasets can be combined with back propagation when there is a need to obtain the three-dimensional distribution of δ in the sample (it is known that δ is approximately proportional to the mass density in the material, see [18]). However we believe that for quantification of a variety of characteristics of cracks and interfaces such as size, spatial density etc., it may be sufficient to reconstruct a 3D image from the Fresnel-propagated shadow images without full phase retrieval. It is the interference fringes that bring out the discontinuities and inhomogeneities, carrying information about the internal structure. Note that the magnified fringes originating from these features are often considered to be an artefact of Fresnel propagation imaging. Yet the spreading of fringes yields a lens-less enlargement of discontinuities, resulting in the ability to visualize micro structural characteristics that are at the edge or below the lateral resolving power of the detector system.

Finally, we note that this technique makes in-situ experiments very feasible. Experiments such as thermal, hydrational or delicate mechanical loading can be performed and structural changes and deformation can be monitored. This can therefore be used as a tool to measure the performance of such biological materials under very low loads, common in physiologically relevant conditions. Such measurements are key to understanding how biomaterials work in the body and they can be performed for many biomaterials, including bone and tendon.