Fresnel-propagated submicrometer x-ray imaging of water-immersed tooth dentin

Simon Zabler; Peter Cloetens; Paul Zaslansky

doi:10.1364/OL.32.002987

X-ray absorption and phase-contrast tomography are important tools for the study of biomaterials. Hard x rays are often used, where biological materials appear partially transparent due to their low absorption so that the distribution of attenuation coefficients and corresponding constituent densities are visualized nondestructively [1]. Specifically for mineralized tissues such as tooth or bone, representative millimeter-sized samples require the use of energies of $20 - 50 keV$ $(λ = 0.2 - 0.6 Å)$ to obtain reasonable radiographs. With phase enhancement [2], coherent imaging allows visualization of micrometer-sized features, which characterize most skeletal tissues.

The structural frameworks of animals that contain mineral, often calcium, are conventionally x rayed when dry. Yet the structural integrity may be compromised because of the abundance of water in the natural state of these materials. For example, bone and similar tissues such as deer antler and dentin contain up to 30% water by volume [3], and they shrink or crack when dehydrated. Precise 3D wet imaging of these materials is presumably important. It is, however, not straightforward, because the x-ray–liquid interactions affect the radiographs and must be accounted for. Both components of the complex refractive index $n = 1 - δ + i β$ are significant for interactions in the $20 - 50 keV$ range (β corresponds to attenuation due to the photoelectric effect and inelastic Compton scattering, δ corresponds to phase retardation due to Thomson scattering [4]). For calcium phosphates β is $(0.5 - 8) \times 10^{- 9}$ , and δ is $(0.5 - 1.5) \times 10^{- 6}$ , whereas for water and protein, the respective values are $(0.6 - 30) \times 10^{- 11}$ and $(1 - 7) \times 10^{- 7}$ . The phase shift is thus much more prominent than the attenuation. In a typical tomography setup (Fig. 1 ), incoming x-ray fields $u_{i n c} (x, y)$ propagate along the optical path in the z direction through and beyond the sample, impinging on a scintillation screen. The screen is imaged onto a CCD array for acquisition. Following a thin lens approximation [5], the x-ray field on a plane $(x, y)$ immediately behind the sample (subscript “0”), ignoring any time dependence $e^{(i ω t)}$ can be described by

u_{0} (x, y) = u_{inc} (x, y) \times T (x, y),

T (x, y) = e^{(π ∕ i λ) \int [1 - n^{2} (x, y, z)] d z} \approx e^{- B (x, y) - i ϕ (x, y)} .

Here

T (x, y)

is a complex transmission function and

ϕ (x, y)

and

B (x, y)

represent the outcomes of propagation through the sample. ϕ and B are related to the real and the imaginary parts of

n (x, y, z)

in different regions of the sample, by line integrals of δ and β along z [2]. The intensity on a radiograph recorded at the sample–detector distance

D = 0

is

I_{0} = ∣ u_{0} ∣^{2} = e^{- 2 B}

at each point

(x, y)

. Thus, only the imaginary component of n contributes to conventional x-ray images.

A substantial increase in detail visibility occurs when phase-shift effects are recorded. This is particularly useful for visualizing interfaces and inhomogeneities in the microstructure of biomaterials, where precise attenuation coefficients are not needed. With monochromatic partially coherent sources, phase enhancement may be induced by various methods [2], the simplest of which suggests increasing D [6], as shown recently in seeds or teeth [7, 8]. Yet under high-flux conditions $(> 10^{10} photons ∕ s ∕ {mm}^{2})$ , x-ray interactions with the sample environment result in degraded images. Specifically, in water-immersed samples, bubbles appear, probably due to the dissociation of $H_{2} O$ molecules, as shown, for example, for ice [9]. The resulting free radicals form bubbles that continuously expand, move, and float up to the surface. Silhouettes of such bubbles are often observed in phase-enhanced images and are also seen with other liquids (e.g., ethanol).

We consider here various experimental aspects of tomography of water-immersed samples of tooth dentin and propose a method for reliable acquisition and reconstruction of high-resolution 3D images of the microstructure. Our results indicate this to be directly applicable to x-ray imaging of many wet biomaterials at submicrometer resolution.

Tomograms are constructed from a series of angularly varied radiographs of the sample in its immediate environment. In the radiographs, stray intensity contributions may be described by noting that $T (x, y)$ [Eq. (1)] represents a combined transmission effect through the vial $(T_{v})$ , water $(T_{w})$ , bubbles $(T_{B})$ , and sample $(T_{S})$ so that $T (x, y) = (T_{V} \times T_{W} \times T_{B} \times T_{S}) (x, y)$ . Following Fresnel-propagation over a constant distance D for each projection angle φ behind the rotation stage, $I_{D} (x, y, φ) = ∣ u_{D} (x, y, φ) ∣^{2}$ is recorded. For weakly absorbing objects and sufficiently smooth phase shifts, an analytical expression of $I_{D}$ can be derived in reciprocal space [6], where “∧” indicates the Fourier transform

{\hat{I}}_{D} (f, g, φ) = δ_{2 D} (f, g) + F_{1} \times \hat{ϕ} (f, g) - F_{2} \times \hat{B} (f, g) .

Here, intensity is a multiplication of contrast transfer functions

F_{1} = 2 \sin [λ D π (f^{2} + g^{2})]

and

F_{2} = 2 \cos [λ D π (f^{2} + g^{2})]

with the Fourier transforms of ϕ and B (convolutions in real space).

δ_{2 D}

is the Dirac delta distribution at the origin of

(f, g)

, conjugate to

(x, y)

[6, 10]. At small distances

F_{1} \sim 2 λ D π (f^{2} + g^{2})

is equivalent to the Fourier transform of a Laplace filter, and consequently edge enhancement occurs with increased D. As a result, terms of

T_{B}

or

T_{V}

in

T (x, y)

containing sharp interfaces are enhanced, obscuring structural details that are related to modulations of

T_{S}

whose phase shift/attenuation is of interest [note that for an ideal image

I_{S}^{(ideal)} \approx ({\hat{F}}_{1} * ϕ_{S}) - ({\hat{F}}_{2} * B_{S})

]. While

T_{W}

is a function of

(x, y, φ)

,

T_{V}

and

T_{B}

additionally vary in time, appearing in only a subset of the projected angles. For a reasonably smooth, clean vial,

T_{V}

will contribute only minimal stray intensities.

T_{B}

on the other hand, causes severe perturbations due to the sharp interfaces between the liquid and gas bubbles, whose silhouettes are projected onto the radiographic images (Fig. 2 ).

The experimentally recorded radiographs are conventionally normalized by subtraction of the CCD dark current $I_{DC} (x, y)$ and division by bright-field images $(I_{B F} (x, y) = ∣ u_{inc} (x, y) ∣^{2})$ so that normalized intensity ratios $I_{D}^{(corr)} (x, y, φ)$ for each projection angle φ are obtained

I_{D}^{(corr)} (x, y, φ) = \frac{I_{D} (x, y, φ) - I_{D C} (x, y)}{I_{B F} (x, y) - I_{D C} (x, y)} .

This normalization is vital for the removal of inhomogeneities in the beam profile and in the detector response. A 3D virtual volume is then assembled by calculating a stack of y planes of the microstructural feature distribution

ρ (x, z) = f [δ (x, z), β (x, z)]

, from lines of

- \log [I_{D}^{(corr)} (x, φ)]

in the projection images. Inversion of the Radon transform

(R)

[11],

- \log (I_{D}^{(corr)}) = R_{ρ} (x, φ) = \iint_{sample} d v d w δ (v \sin (φ) + w \cos (φ) - x) ρ (v, w) .

[

(v, w)

coordinates of each rotating sample slice] is calculated by standard filtered backprojection so that approximations of

ρ (x, z)

are obtained. As shown elsewhere [12] the phase retardation is incorporated into

ρ (x, z)

and corresponds to the microstructure densities and constitution. Large contributions from stray intensities of

- \log [I_{D}^{(corr)} (x, φ)]

must thus be minimized. Despite appearing in only a limited interval of angular projections,

T_{V}, T_{B}

variations dominate over intensity differences due to

T_{S}

. Discontinuities [13], most notably bubble edges, increase

T_{B}

and extreme dark or bright contributions result in severe perturbations of the 3D image, because they are integrated into many

ρ (x, z)

points.

To reduce the spurious intensities of the moving bubbles $T_{B}$ we exploit the fact that they vary randomly over time. In a sequence of repeated tomographic acquisitions of a given sample recorded at different times, the overlap between extreme $T_{B}$ contributions is quasi negligible. With a set of N (preferably odd) identical projections from different acquisition runs, each offset by a small vertical sample shift $Δ y_{j}$ but recorded at the same projection angle φ, one obtains a series of images $I_{D}^{(j)} (x, y - Δ y_{j}, φ)$ . After normalization, the images $j = 2 \dots N$ are aligned with $I_{D}^{(1)}$ and time-median images are produced, whereby each projection image point contains only median values. By choosing sufficiently long time intervals $Δ t = t_{j + 1} - t_{j}$ , improved estimates of $T_{S}$ may be obtained, while removing the destructive effects of $T_{B}$ .

The above approach was used to image micrometer-sized tubules in small samples of tooth crown dentin $(1 mm \times 1 mm \times 4 mm)$ . Samples were immersed in water within poly(methyl methacrylate) (PMMA) vials. The samples, from several different teeth, were arranged such that remaining small fragments of the enamel (which normally coats the external surface of teeth) were situated on the upper edge, assisting in orientation. Each sample and vial were placed at the center of a high-precision rotation stage of the partially coherent high-flux x-ray imaging beam line ID19 in the European Synchrotron Radiation Facility (ESRF), Grenoble, France. The x-ray detector ( $17 μ m$ thick LAG:Eu scintillator, $10 \times ∕ 0.40$ Olympus lens, $2 \times$ eyepiece, and FRELON $2 K$ CCD camera) was positioned at one of several distances $(D = 10 - 200 mm)$ behind the sample (Fig. 1).

In uncorrected reconstructed slices of the water-immersed dentin [Fig. 2a, $D = 100 mm$ , x-ray energy $E = 22 keV$ ] white streaks affecting large regions in the 3D reconstructed image are seen, caused by stray intensities induced by bubbles [inset Fig. 2b]. Improved reconstructions [Fig. 2c] were produced from time-median images [Fig. 2d], created from the median values of matching radiograph points from three sequential sets and recorded at identical sample projection angles $(Δ t = 20 \min)$ . Artifacts [Fig. 3a ] are significantly reduced in the improved reconstruction [Fig. 3b], revealing the outlines of the tubules and variations in the density and 3D orientations. In particular, differences in δ due to variations in thickness and projected electron density $ρ_{e l}$ of the $0.5 - 4 μ m$ mineral sheaths surrounding the tubules are seen (note that $δ \sim λ^{2} ρ_{e l}$ [4]). With increased propagation distance [ $D = 180 mm$ , inset Fig. 3c] fringes around the tubules appear broader, yet the tubule center-to-center distances remain unchanged [8]. To the best of our knowledge, no other form of nondestructive (whole tissue) microscopy reveals such 3D structural detail in similar biomaterials.

We conclude that most time-dependent stray intensity contributions can be removed from projection images of water-immersed biomineralized samples. Stray contributions, attributed mainly to the formation of gas bubbles degrade the visibility of micrometer-sized features that characterize skeletal tissues. During such experiments, wet samples do not undergo dehydration, and therefore it is possible to acquire large numbers of radiographs repeatedly. When additionally small deliberate shifts to the sample stage are induced, artifacts caused by detector components are reduced, leading to additional improvement in the visibility of features in the microstructure. In samples of water-immersed dentin, improved visibility of interference fringes resulted in detailed representation of the $0.5 - 1 μ m$ sized tubules and peritubular sheaths in the tomograms. Additional structural insights may arise by exchanging water for other liquids (ethanol, saltwater) such that diffusion effects may be tracked. It is thus believed that this form of nondestructive liquid-immersed 3D microscopy has a wide application, far beyond tooth structure imaging.

The authors thank John Currey and Peter Fratzl for discussions and support.

Fig. 1 Component setup for Fresnel-propagated microtomography: images of the liquid immersed sample are acquired at several detector positions.

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Fig. 2 Projection radiographs and tomographic slices of wet dentin samples: (a) Reconstructed slice of Fresnel-propagated radiographs: note white streaks, which are due to stray intensity contributions originating from bubbles in the water. (b) Radiograph of water-immersed dentin (dense enamel on upper area appears dark) contains high-contrast bubble silhouettes. (c) Slice in a 3D time-median reconstruction with fewer stray intensities. Structural detail is observed (see Fig. 3). (d) Typical time-median radiograph.

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Fig. 3 Microstructural detail in dentin (a) high- and low-density regions of mineral in the dentin are obscured by image noise in slices of tomograms produced from conventional radiographs. The combination of ring artifacts and stray intensity contributions makes visualization and analysis difficult. (b) Improved reconstructed images, produced from three sequential scans, result in high-quality virtual microscopy images of interference rings corresponding to wet $1 μ m$ thick tubules $(D = 100 mm)$ . Images obtained for larger propagation distances (c) ( $D = 180 mm$ ) show enlarged tubules, yet the center-to-center distances $(8 - 10 μ m)$ remain constant.

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Fresnel-propagated submicrometer x-ray imaging of water-immersed tooth dentin

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Optics Letters