1. Introduction

Conventional X-ray imaging and computed tomography (CT) systems rely on the attenuation of X-rays for contrast. Imaging weakly absorbing materials, e.g., of biological nature, results in particularly low internal contrast, which can effectively limit the applicability of X-ray imaging and CT for such samples. In addition to being absorbed, the X-ray wavefront interacts with the sample by being phase shifted [1], the cross section for which can be up to three times larger than that of attenuation [2]. Since the 1960s, X-ray phase contrast imaging and (later) CT were introduced as a possible solution to the weak contrast created by low-absorbing materials. Several established methods that convert phase shifts into measurable intensity differences exist, including crystal interferometry [3], grating interferometry [4], analyzer-based imaging [5], speckle-based imaging [6], grating-based non-interferometric imaging [7], and propagation-based imaging [8].

The simplest phase-contrast arrangement, which, in fact, requires the same components as conventional X-ray imaging and CT, with an additional requirement for spatial coherence, is propagation-based imaging (PBI - also termed in-line phase contrast imaging). Briefly, phase variations are transformed into intensity variations via free-space propagation of the X-ray wavefront from the object to the detector; edge enhancement by near-field interference fringes (Fresnel diffraction) is created around the object’s edges and interfaces. Since this method does not yield a direct measurement of the phase, a phase retrieval approach is required; this provides a quantitative relation between the measured phase-contrast (i.e., intensity variations) and the characteristics of the sample [9]. The need for spatially coherent X-ray beams means PBI is widely used at synchrotrons [10,11], however it has long been known that its laboratory implementation is possible with polychromatic X-ray microfocus sources [1], although at a lower flux (and therefore longer exposure times) c.f. synchrotron sources.

Various studies have been performed on purpose-built PBI laboratory systems. These encompass hardware advances [12,13], development of phase-retrieval algorithms [9], as well as various application-oriented studies in e.g. biomedicine and material science. Previous studies have also examined the characteristics of commercially available X-ray scanners and investigated their suitability for propagation-based phase contrast imaging, for example using the ZEISS Xradia 500 Versa [14]. The geometry of the system was optimized by varying the total source-to-detector distance at constant magnification and varying the magnification at constant source-to-sample distance, while imaging test samples [14]. The same system was subsequently used to image composite materials and biological samples and improvement in contrast was proven through application of phase-retrieval [15]. ZEISS Xradia 520 Versa systems were also optimized by varying the magnification for two different source-to-detector distances [16] and in terms of source-to-sample distance, magnification, source size, and anode potential [17]. Finally, An EasyTom XL Ultra 230-160 micro- and nano-CT scanner was also investigated for PBI [18]; the physical setup, imaging geometry and phase retrieval parameters were all optimized to improve the visibility and the contrast-to-noise ratio (CNR) of low-attenuating materials.

The aim of this work is to optimize a laboratory setup, confined to a fixed-length equal to the length of commercial scanners from Nikon Metrology (Tring, UK) and investigate the implementation of PBI as a routine procedure, additional and complementary to conventional X-ray imaging and CT. The feasibility of PBI was studied in a laboratory system with the following characteristics: fixed source-to-detector distance of ∼90 cm, a commercial polychromatic microfocus X-ray source with a variable focal spot size, and a commercial flat panel detector with a 50 µm pixel size.

2. Methods

2.1 Experimental setup

The laboratory setup is schematically depicted in Fig. 1. A Hamamatsu microfocus source with a Tungsten (W) anode (model L12161-07) was utilized. The source was operated at various settings to investigate their effect on the optimization of the imaging setup. Specifically, six different settings were explored, in terms of focus mode, tube voltage, and tube current; these are shown in Table 1. In all cases, the maximum tube current compatible with the used kV at each source setting was used. The source shape was characterized from the line spread function (LSF) measurements using an edge, with a Cobalt (Co) thickness equal to 9.5 mm, positioned at 0° and 90° rotation [19] and found to be well approximated by a Gaussian with 12 µm and 50 µm full width at half maximum (FWHM) for small and medium focus mode, respectively. Prior to any acquisitions, the source was allowed to stabilise for 2 hours. The sample stage consisted of a Newport linear stage (model M-ILS150BPP) for the positioning of the sample along the X-ray beam propagation (i.e., along the z axis) and a Newport rotation stage (model SR50CC) for the angular rotation. The source-to-detector distance, R_tot, was constant at 919 mm. The detector was a Hamamatsu CMOS-based flat panel sensor (model C9732DK) with 2368 (h) × 2340 (v) 50 × 50 µm² pixels. Its point spread function (PSF) was measured to be a Gaussian with 170 µm FWHM along both directions (see Supplement 1), from the LSF measurements of an edge of a Fluke Biomedical X-ray test pattern (model number 07-521), with a Pb thickness equal to 0.05 mm. The source-to-sample distance, R1, was varied between 50 mm and 320 mm, with a step size of 15 mm. The smallest R1 investigated was determined by the physical limitations of the system, as the sample could not be placed closer than 50 mm from the focal spot.

 

Fig. 1. Schematic diagram of the laboratory propagation-based phase-contrast imaging setup.

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Table 1. X-ray source settings investigated.

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2.2 Data acquisition

Initially, planar images were acquired to identify the sample-to-detector (i.e., propagation) distance R2 at which the projection images would exhibit the maximum edge enhancement; note that in our study this automatically determines source-to-sample distance R1 and magnification M. A sample with multiple features was created and used in all cases; it consisted of a tissue-engineered esophageal scaffold [20] (wrapped in paper), a 1 mm diameter PTFE wire, and a 0.4 mm diameter PET wire. The scaffold was inserted in a 6 mm diameter plastic straw. For both small and medium focus modes, the tube voltage was set to 60 kVp, and a series of sample images were acquired as a function of the source-to-sample distance R1, which was varied from 50 mm to 320 mm with a step size of 15 mm (19 positions in total). For each setting, 10 dark and 10 flat images were acquired prior to the sample projection images.

Once the propagation distance corresponding to maximum edge enhancement is determined for both focus modes, step-and-shoot CT scans were acquired. The same sample was used, and all six settings shown in Table 1 were investigated. 720 projections were acquired from 0° to 359.5° with a 0.5° step with a 0.5 s exposure time. Following spatial resolution optimization, a final scan was acquired to investigate the performance of the scanner on a larger biological sample, a freeze-dried rat heart (kept at room temperature during the scan). The rat heart was placed in a 15 mm diameter plastic cylinder. The number of projections and exposure time were the same as before, however, this scan was implemented as a continuous scan (flyscan), resulting in an overall scan time (= exposure time) of 360 s.

2.3 Simulations

The experiment, image acquisition of a wire, was initially modelled using a 1D wave-optics simulation. The model used, described in detail in Ref. [21], simulates the free-space propagation of X-rays between two parallel planes using the Fresnel-Kirchoff theory of diffraction in the Fresnel approximation; the complex amplitude of the electric field at the latter plane is calculated as the convolution between the complex amplitude of the electric field at the initial plane and the Fresnel propagator. The sample is modelled as complex transmission function and the detected signal is calculated by taking into account the X-ray source distribution and the detector PSF through additional convolutions; each function is sampled along the x direction (horizontal), at a 1/Δ_x sampling rate.

The experimental setup presented in Fig. 1 was simulated, with the sample being a 1 mm diameter PTFE wire at variable distances, R1, from the source, equal to the experimentally investigated ones. A monoenergetic X-ray source of 17.4 keV (corresponding to the mean energy of the source at 60 kVp) was simulated. A sampling step of Δ_x = 50 nm was used to sample all functions. A total of 19 planar images of the sample were acquired, one for each investigated R1, for both source sizes (small focus mode: 12 µm FWHM and medium focus mode: 50 µm FWHM). Each of the simulated images included 2340 profiles along the vertical direction, to replicate the size of the experimentally acquired planar images. Initially, noise-free images were simulated, to quantify the relative fringe amplitude (please see Section 2.5 Data processing) of the simulated images. Then, Poisson noise was added to the simulated images to investigate the noise propagation from the simulated images to the retrieved images, as a function of source size and R1. The number of photons per pixel assumed for the added Poisson noise for the small (N = 2 × 10³) and for the medium (N = 6 × 10³) source size was estimated by matching the experimentally observed standard deviation of the background counts (i.e., noise) for each source size at 60 kVp with that of the simulated images. The correlation in the experimentally observed noise resulting from the detector blur was modelled in the simulations by convolving the 2D noisy projection images by the 2D detector PSF.

2.4 Theoretical considerations underpinning data analysis

Phase contrast is defined as the image contrast arising from the phase variations of the X-ray wave-field [22]. The phase contrast image formation is described using the Fresnel propagator [22]. For simplicity, the following assumptions are made, 1) a thin (along the z direction) and weakly absorbing sample, 2) a parallel beam geometry, and 3) a monochromatic X-ray source of wavelength, λ.

The intensity distribution directly after a sample, z = R1, of refractive index $n\; = 1 - \; \delta + i\beta $, where $\delta $ is the real decrement from unity responsible for changes in phase and $\beta $ is the imaginary component responsible for attenuation, is given by the absorption of the sample,

The intensity distribution being recorded at a propagation distance R2, z = R2, is then obtained by also considering the refraction effect generated by the phase shift, Φ$({x,y} )$,

Equation (3), which is a form of the ‘transport-of-intensity equation’ [22], relates the intensity distribution being recorded at a propagation distance R2 to both the intensity (I_sam) and the phase (Φ) variations of the field over the plane z = R1. The second term on the right side of Eq. (3) represents the local variation, either increase or decrease, in intensity which results from the local curvature of the phase in a given plane z (e.g., at R2); a positive wave-front curvature leads to a decrease in the propagated intensity, while a negative wave-front curvature leads to an increase in propagated intensity. This local redistribution of optical energy upon propagation results in edge enhancement and in the creation of the so-called fringes. Equation (3) shows that this contrast, the phase contrast, is proportional to the intensity of the fringes, and thus is proportional to the propagation distance R2 and is strongest where abrupt changes in δ occur, i.e., at the edges of sample details. Maximizing the fringes is thus a major part of the optimization process for a propagation-based phase-contrast imaging system. However, as discussed below, the relationship between fringe intensity and R2 is not monotonic when the effect of the finite source size and the detector PSF is considered.

The above theory can be extended to a polychromatic beam and a cone-beam geometry. This is done by considering a wavelength λ_eff, corresponding to the effective energy of the X-ray beam and the geometric magnification factor,

In this case, images (both planar and the reconstructed CT ones) are re-scaled, and the pixel size at the sample plane reduces from the detector pixel size ps (= 50 µm) to:

Additionally, the propagation distance becomes the effective (also termed “defocusing”) propagation distance,

The edge enhancement can be quantified using the relative fringe contrast,

Next, detector PSF and finite source size must be taken into account, as they both affect both spatial resolution and fringe contrast [14] beyond the effects of attenuation and refraction predicted by Eq. (3). Their effect, assuming a Gaussian distribution, is included through their standard deviations (σ_s for source distribution and σ_det for detector PSF). The imaging system’s spatial resolution at the sample plane [14,23], quantified by the standard deviation, σ_sys, is given by:

The magnification value yielding best spatial resolution M_{opt_res} = 1 + ${\left( {\frac{{{\sigma_{det}}}}{{{\sigma_s}}}} \right)^2}$, and the optimal resolution itself ${\sigma _{min}} = \; \sqrt {\frac{{{{({{\sigma_s}{\sigma_{det}}} )}^2}}}{{({\sigma_s^2 + \sigma_{det}^2} )}}} $, can both be calculated by re-arranging Eq. (9). This shows that neither M_{opt_res} nor ${\sigma _{min}}$ depend on the total source-to-detector distance R_tot.

The effect of source distribution and detector PSF on fringe contrast is less straightforward. Dierks & Wallentin [16] deduced an expression for fringe contrast as a function of the source-to-detector distance, R_tot, and magnification M_geom, in which fringe contrast increases as R_tot is increased. For constant R_tot, it is maximized at magnification:

Thus, the magnification resulting in maximum fringe contrast does not depend on R_tot; however, two aspects should be noted: the value of the maximum fringe contrast is itself a function of R_tot (and increases with increasing R_tot as long as near-field conditions are respected), and the magnification at which the fringe contrast is maximised is different to that optimising spatial resolution (apart from the special case σ_det = σ_s).

Nesterets et al. identified a trade-off in the optimization of the geometry of a laboratory-based PBI system [12], which is worth mentioning here. They show how the contrast of a blurred (non-ideal) edge increases with increasing source-to-detector distance R_tot, whereas the signal-to-noise ratio (SNR) reduces; SNR is proportional to the contrast and inversely proportional to R_tot [12].

2.5 Data processing

The experimental planar images acquired as a function of R1 (and therefore M_geom) were dark and flat-field corrected. Then, a ROI was selected to encompass the PTFE wire and part of the background along the horizontal (x axis) direction. Multiple profiles along the vertical (y axis) direction corresponding to 250 µm height were used to quantify the relative fringe amplitude at each magnification. The profiles were normalized to the mean intensity of the background. Both the right and the left edge of the wire were used to measure the fringe amplitude. The mean and standard deviation of the relative fringe contrast at the left and right edges of the wire of all profiles were calculated for each source setting, as a function of R1. The same procedure was used to quantify the relative fringe amplitude of the simulated (noiseless) PTFE wire profiles at different R1 distances.

It should be noted that a phase-contrast image contains the maximum phase shift information at a certain propagation distance, R2, depending on the setup. This arises from the Fourier Transform of the wave function which is proportional to the optical transfer function (OTF) for Fresnel diffraction [24]. Considering our assumptions of a thin (along the z direction) and weakly absorbing sample, the real and imaginary parts of the OTF for Fresnel diffraction is derived to describe the amplitude and phase contrast, respectively, both being functions of u^’ = (λz)^1/2u. The phase component of the OTF reaches its first maximum at u^’ = (1/2)^1/2 (while the amplitude component is 0) [25], thus, it is maximized at the optimal spatial frequency,

We used the single-distance phase-retrieval algorithm developed by Paganin et al. [26] with both the experimentally acquired and simulated planar images, which extracts quantitative phase shifts for a homogeneous sample. The assumption of a homogeneous sample means that the ratio ${\gamma _0}({{\lambda_{eff}}} )= \frac{{{\delta _0}}}{{{\beta _0}}}$ at the effective energy of the X-ray beam λ_eff is constant within the sample. This being the case, attenuation (Eq. (2)) and phase shift (Eq. (4)) can be retrieved from a single propagation-based phase-contrast image. By defining the Paganin filter as:

Following retrieval of both the experimentally acquired and simulated planar images (with Poisson noise), the CNR was quantified,

CT reconstruction of both raw (edge-enhanced attenuation) and retrieved experimental images was performed with a GPU implementation of the Feldkamp-David-Kress algorithm [33] for cone-beam reconstruction using the ASTRA toolbox [34,35]. The reconstructed volumes had a voxel size equal to ps_eff³. The spatial resolution was estimated from the edges of the PTFE wire at the axial CT slices. Error functions were fitted to ≥ 20 edges along the x and z directions. Their derivatives were then computed to obtain LSFs, which enabled extracting their FWHM. The mean and standard deviation of the extracted spatial resolution values at each direction and for each CT scan were then computed.

3. Results and discussions

3.1 Theoretical calculations

Geometric magnification factor M_geom and effective pixel size ps_eff were calculated as a function of the source-to-sample distance R1; these are shown in Fig. 2. M_geom ranged from 2.9 (17.4 µm effective pixel size) to 18.4 (2.7 µm effective pixel size). As a result, the effective field-of-view (FoV) ranged from 6.4 mm × 6.4 mm at 50 mm source-to-sample distance to 40.7 mm × 41.2 mm at 320 mm source-to-sample distance.

 

Fig. 2. Magnification factor (filled circles) and effective pixel size in µm (open diamonds) as a function of the source-to-sample distance R1.

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It is important to also note the variation of the effective propagation distance Z_eff as a function of R2 (and of R1, since R1 = R_tot – R2) for a system with constant R_tot; this is shown in Fig. 3. At the sample position closest to the source (50 mm) the sample-to-detector distance (i.e., propagation distance, R2) was greatest (869 mm), however the effective propagation distance was the smallest (= 47 mm). The largest effective propagation distance experimentally investigated here was 209 mm, which occurred at 599 mm of propagation distance. A linear relationship between propagation distance and effective propagation distance is only achieved when a change in R2 is accompanied by a proportional change in R1, so as to keep a constant magnification. For a system with a constant R_tot, the effective propagation distance approaches zero (i.e., zero phase-contrast) for both M_geom = 1 (R2 = 0) and M_geom = ∞ (R1 = 0). The maximum effective propagation distance in the reported system (which, however, was not experimentally investigated here) is 230 mm at R1 = R2 = z/2 = 460 mm.

 

Fig. 3. Effective propagation distance Z_eff as a function of sample-to-detector distance R2 for the whole range; the experimentally investigated range is highlighted by red crosses.

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The final theoretical calculations were to estimate the source-to-sample distances that maximize fringe contrast and spatial resolution; these are presented in Fig. 4 as a function of the source FWHM (see Table 2). Considering a detector PSF of 170 µm FWHM and a source distribution of 12 µm FWHM, the fringe contrast should be maximum at R1 = 60.6 mm (M_geom = 15), and the spatial resolution at R1 = 4.6 mm (M_geom = 202). For a 50 µm FWHM source, these values are 208.9 mm (M_geom = 4.4) and 73.2 mm (M_geom = 12.6) for fringe contrast and spatial resolution, respectively.

 

Fig. 4. Optimum source-to-sample distance R1 as a function of source size for maximizing the fringe contrast (blue diamonds – solid line) and spatial resolution (black circles – solid line). The vertical lines at 12 µm (dashed line) and 50 µm (dotted line) correspond to the experimentally investigated cases.

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Table 2. Summary of results for the two source settings.

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3.2 Radiography

Here, the sample-to-detector distance R2 (and thus R1) at which 2D images exhibit maximum edge enhancement for both focus modes was identified, using both experimental and simulated results, in order to inform the optimization of a fixed-length micro-CT system for propagation-based phase-contrast imaging. Edge enhancement was quantified using unretrieved images of the PTFE wire at different source-to-sample distances R1, and propagation distances R2. Example profiles for two source settings, at the theoretical optimum R1 for fringe contrast (50 mm for the small and 185 mm for the medium focal spot size, respectively) are shown in Fig. 5. The simulated (noiseless) profiles along the PTFE wire for the same source settings and source-to-sample distances are also shown. The fringe width and intensity varied with the source size and effective propagation distance, as can be seen in Fig. 5 (and below, in Fig. 6). The fringes observed with the medium focal spot were wider compared to those with the small focal spot, as expected considering that the width of the image edge is proportional to the system’s spatial resolution σ_sys [36].

 

Fig. 5. Experimentally measured (black solid lines) and simulated (red dashed lines) normalized intensity profiles of unretrieved images across the 1 mm thick PTFE wire with (a) a small X-ray focal spot (at 60 kVp and 166 µΑ) for a source-to-sample distance of 50 mm and (b) a medium X-ray focal spot (at 60 kVp and 500 µΑ) for a source-to-sample distance of 185 mm.

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Fig. 6. Experimentally measured (circles) and simulated (dashed lines, fringe contrast as a function of source-to-sample distance R1 for a small X-ray focal spot (black line/symbols) and a medium X-ray focal spot (small line/symbols). The lines are guides for the eye. The error bars represent the combination of the calculated standard deviation of the relative fringe contrast of all profiles and that of the background.

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The quality of the phase-contrast images was then assessed taking into account the optimal spatial frequency (Eq. (11)) for the reported setup, similarly to [14]. Considering an effective energy of the X-ray beam of ≈17 keV and the effective propagation distances investigated here (Fig. 3), the spatial frequency u_opt yielding the maximum of the OTF was calculated to reduce from 3.8 × 10⁵m^-1 at R1 = 50 mm to 1.8 × 10⁵m^-1 at R1 = 320 mm. The effective pixel size (Fig. 2) corresponded to spatial resolutions of 1.8 × 10⁵m^-1 at R1 = 50 mm, reducing to 0.3 × 10⁵m^-1 at R1 = 320 mm. Thus, the optimal phase contrast was observed in the images with the shortest R1, where the inverse of the optimal spatial frequency was closest to the effective pixel size at the corresponding sample plane.

The relative fringe contrast describing the edge enhancement was quantified for the two focal spot sizes as a function of R1 using Eq. (8); this is presented in Fig. 6 for both the experimentally measured and simulated (noiseless) planar image acquisition (see Table 2 for a summary). The experimental relative fringe contrast at the position closest to the detector (R1 = 320 mm) was 0.012 ± 0.008 and 0.017 ± 0.009 with the medium and the small focal spots, respectively. With increasing propagation distance, the experimental fringe contrast increased, reaching a value of ≈0.015 with the medium focal spot between R1 = 50 mm and R1 = 275 mm (within error bars). The experimental fringe contrast with the small focal spot kept increasing with increasing propagation distance, reaching a maximum at R1 = 95 mm and remaining constant within uncertainties from there to the smallest R1; it was measured to be 0.050 ± 0.008 at R1 = 95 mm and 0.050 ± 0.009 at R1 = 50 mm. The relative fringe contrast as was quantified from the simulations agreed with the experimental one, within uncertainties. The optimum source-to-sample distance for maximizing fringe contrast as identified from the simulations, was 215 mm for the medium focal spot and 65 mm for the small focal spot; these values matched the calculated one using Eq. (10) (shown in Fig. 4).

An example of an acquired image and its corresponding retrieved projected thickness for the 1 mm diameter PTFE wire is shown in Fig. 7. Profiles along the sample are also shown in the same figure. By comparing the profiles of the acquired and retrieved images, it can be observed that the phase-retrieval increases the SNR while preserving the sharpness of the image to a reasonable extent. An effective X-ray energy of 17.4 keV was used, which yielded a thickness value corresponding to the nominal one (1 mm).

 

Fig. 7. Acquired (a) and retrieved projected thickness (b) images of the 1 mm diameter PTFE wire, obtained using a small focal spot (at 60 kVp and 166 µΑ) at a source-to-sample distance of 50 mm. The profiles along the red dashed line in (a) and (b) can be seen in (c) and (d) respectively (scale bar = 200 µm).

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The CNR of the PTFE wire in the retrieved planar images was calculated, as described in §2.5, to investigate the effect of the Paganin retrieval to the PTFE wire contrast and to the background noise, as functions of the source-to-sample distance, R1. The results are presented in Fig. 8 (see Table 2 for a summary). The Paganin retrieval corresponds to the application of a low-pass filter (specific to the sample) to the acquired images. The γ₀ value used for the retrieval was 837 (effective energy of 17.4 keV). The extracted values of PTFE at 17.4 keV using xraylib were β₀ = 1.7318 × 10⁻⁹ and δ₀ = 1.4502 × 10⁻⁶.

 

Fig. 8. CNR analysis of the retrieved images as a function of R1, using the small (a)-(c) and the medium (d)-(f) X-ray focus mode. The PTFE wire contrast ((a) and (d), black squares), the background noise ((b) and (e), black diamonds), and the CNR ((c) and (f), black circles) for each focus mode is shown as a function of R1. The background noise of the simulated planar images (open red circles) is also shown. The red lines at 1000 µm retrieved thickness are guides for the eye.

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The retrieved thickness of the PTFE wire (Eq. (13a) was calculated to be 1000 µm (within noise) for both source sizes and at all investigated values of R1. However, the background noise was found to vary with both source size and R1. The background noise in the retrieved thickness images using the small focal spot increased from 3.7 µm at R1 = 50 mm to 8.2 µm at R1 = 320 mm. Similarly, the background noise in the retrieved thickness images using the medium focal spot increased from 2.2 µm at R1 = 50 mm to 5.5 µm at R1 = 320 mm. The background noise of the retrieved simulated images as function of R1 was also quantified; it increased from 3.5 µm at R1 = 50 mm to 8.6 µm at R1 = 320 mm with the small focal spot and from 2.1 µm at R1 = 50 mm to 5.4 µm at R1 = 320 mm with the medium focal spot.

The effect of phase retrieval on noise in projection images has been mathematically described by Nesterets et al. [37] as being determined by a factor A: the larger this factor A, the larger the noise reduction due to phase retrieval. The factor A depends on the refractive properties of the sample, on the X-ray energy, on the geometry of the setup, and on the detector pixel size [38]. This factor is further simplified when considering the same material (here, the background of the wire sample), a constant X-ray photon energy, the same detector, and a constant source-to-detector distance; under these conditions, A ∝ R2 × M_geom. In this case, an increase in the sample-to-detector distance, R2 (equally, a reduction in R1) and/or an increase in geometric magnification, M_geom, results in an increase of the factor A which, in turn, determines a larger noise reduction due to phase retrieval [37]. This matches our observations, presented in Fig. 8; a lower noise is presented in the retrieved images at small R1, which increases as R1 increases. It has also been shown that the variance of random noise in the acquired images (assuming that it satisfies Poisson statistics) is propagated to the corresponding retrieved images [37]. As such, the observed variation of noise in the retrieved images with the source size can be attributed to different background noise in their corresponding acquired images; the X-ray tube power with the medium focal spot was 3× larger c.f. the small focal spot (since the X-ray tube voltage was the same in both cases, and equal to 60 kVp, whereas the X-ray tube current was 166 µΑ for the small focal spot and 500 µΑ for the medium focal spot), leading to a correspondingly larger number of incident photons per pixel (and thus, smaller noise).

The resultant CNR of the PTFE wire in the retrieved planar images increased with decreasing R1 and with increasing photon flux, in line with the above considerations. More specifically, the CNR decreased from 270 at R1 = 50 mm to 122 at R1 = 320 mm with the small focal spot and from 455 at R1 = 50 mm to 180 at R1 = 320 mm with the medium focal spot. Example profiles of retrieved images using the small focal spot (at 60 kVp and 166 µΑ) and the medium focal spot (at 60 kVp and 500 µΑ) at the minimum (50 mm) and the maximum (320 mm) investigated R1 values can be found in Supplement 1, Fig. S2.

3.3 Computed tomography

Following the phase-contrast (edge enhancement) optimization from the acquired radiographs for both source focus modes, step-and-shoot CT scans were acquired to aid the spatial resolution optimization of the system. Three CT scans were obtained for each investigated source setting (see Table 1). These were at the three smallest R1 values of 50 mm, 65 mm and 80 mm for the small source, with the choice being based on fringe contrast peaking at small source-to-sample distances (Fig. 6) and spatial resolution being optimal at R1 = 4.6 mm (Fig. 4). For the medium focal spot size, the three CT scans were obtained at R1 values of 185 mm, 200 mm, and 215 mm R1, considering the stable behavior of the fringe contrast between R1 = 50 mm and R1 = 275 mm (Fig. 6).

Since the sample consisted of multiple materials, phase retrieval was performed for each material separately. First, the projected thickness of the 1 mm diameter PTFE wire was retrieved from the flat-field corrected projections. In this case, the nominal value of γ₀(λ_eff) was used. Phase was extracted from the projected thickness calculations by using Eq. [13c], followed by CT reconstruction. Examples of reconstructed axial planes, without and with phase retrieval optimized for the 1 mm diameter PTFE wire, are shown in Fig. 9 alongside the corresponding PTFE wire profiles (small focal spot, 60 kVp, 166 µA, and R1 = 50 mm). The non-retrieved slice was noisy and, although enhancing fringes appeared at the edges of the PTFE wire, the attenuation signal of the biological sample was swamped by the noise. The retrieval significantly increased the contrast of both the PTFE wire and the biological sample while suppressing the noise. Cupping artefacts from beam hardening were visible in the retrieved image.

 

Fig. 9. Reconstructed acquired (a) and retrieved (b) images, with retrieval optimized for the 1 mm diameter PTFE wire. The profiles along the red dashed lines in (a) and (b) can be seen in (c) and (d) respectively. (e) and (f) shows zoom-ins on the unretrieved (e) and retrieved (f) images, with retrieval optimized for the biological sample, showing their respective full grayscale range. (Scale bar = 1 mm).

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The lack of a priori knowledge on the biological sample in terms of its γ₀(λ_eff) (assuming a uniform material) made applying Paganin retrieval not straightforward. Instead, the approach presented in [31] was used. Initially, a phase-retrieved interface between the biological sample and air in a CT slice was obtained by applying the Paganin phase retrieval algorithm with a series of γ₀ values. Then, an error-function-based model was fitted to the interface and the fit parameters were used to identify the correct phase retrieval. A value of γ₀ = 1000 prevented both under-smoothing and over-smoothing effects in the retrieved image. A reconstructed axial slice of the biological sample retrieved with γ₀ = 1000 is shown in Fig. 9. The same non-retrieved slice is also shown for comparison purposes. The increased contrast and reduced noise in the retrieved slice are apparent.

Following retrieval (optimized for the PTFE wire) and reconstruction of all 18 acquired CT scans, the spatial resolution was quantified as described in §2.5. The reconstructed axial, sagittal, and coronal planes of the scans with the best spatial resolution for the small and medium focal spot size are shown in Fig. 10, along with the corresponding line spread functions extracted from the PTFE wire edges (see Table 2 for summary). The best spatial resolution achieved using the small focal spot was (14 ± 1) µm and (16 ± 1) µm along the x and the z axis respectively, at R1 = 50 mm. Similarly, the best spatial resolution achieved using the medium focal spot was (53 ± 4) µm and (51 ± 3) µm along the x and the z axis respectively, at R1 = 200 mm. Despite the better spatial resolution when using the small focal spot compared to the medium focal spot, it is worth mentioning the difference in their corresponding FoV. This was 6.4 × 6.4 mm² for the small focal spot (at 50 mm R1) and 25.5 × 25.8 mm² for the medium focal spot (at 200 mm R1). As such, the best spatial resolution with the reported system can only be achieved when imaging samples up to 5.7 × 5.8 × 5.8 mm³ (h × l × w).

 

Fig. 10. Reconstructed axial (a), (e), sagittal (b), (f), and coronal (c), (g) planes and the corresponding line spread functions (d), (h) extracted from the edges of the PTFE wire along the x and z axis, shown for small focal spot scan (60 kVp, 166 µA) at R1 = 50 mm (a)-(d) and medium focal spot scan (60 kVp, 500 µΑ) at R1 = 200 mm (e)-(h).

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The resolution estimates for all 18 CT scans are shown in Fig. 11. The mean and standard deviation of the in-slice values along the x and z axis were combined, resulting in a single mean with its associated standard deviation. Eq. (9) was used to calculate the system’s spatial resolution for an R1 range from 50 mm to 320 mm (M_geom = 4.7), with the results also shown in Fig. 11. Variations of spatial resolution with source voltage and current were not observed. Instead, the spatial resolution varied with the focal spot size and the source-to-sample distance. The estimated spatial resolution values agreed with the theoretically calculated ones within uncertainties. The results of Fig. 11 suggest that the optimization of the geometry of the system in terms of spatial resolution was more crucial for the small focal spot. The spatial resolution with the small focal spot was calculated to increase from 15 µm at R1 = 50 mm to 60 µm at R1 = 320 mm, whereas the spatial resolution with the medium focal spot was found to range between 49 µm to 68 µm across the whole investigated R1 range, with a minimum at R1 = 70 mm.

 

Fig. 11. Spatial resolution estimates extracted from the edges of the PTFE wire in the reconstructed axial slices, plotted as a function of the source-to-sample distance R1 for three different X-ray tube voltages (and corresponding currents, see Table 1) and for scans with the small and the medium focal spot sizes. The spatial resolution calculated using Eq. (9) for the small (dashed line) and medium (dotted line) focal spot is also shown.

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Finally, a CT flyscan of a rat heart was acquired. Considering the results of the relative fringe contrast (Fig. 6) and the spatial resolution (Fig. 9), the small focal spot was selected. The size of the sample imposed a minimum source-to-sample distance of 200 mm, with smaller distances resulting in an insufficient FoV. At this magnification, the voxel size was (10.8 µm)³ for a volume of approximately 25 mm in all directions. The system’s spatial resolution expected from Eq. (9) was 38 µm. The γ₀ value (assuming a uniform material) used in the Paganin retrieval, determined on the basis of providing the best image quality, was 1200. Different γ₀ values were trialled in turn, and the reconstructed images were assessed from a qualitative point of view; under- and over-retrieved images suffered from low CNR and poor spatial resolution (over-smoothed edges), respectively. A value of 1200 gave the best reconstructed image in terms of trading-off CNR and spatial resolution. Reconstructions without and with Paganin retrieval are shown in Fig. 12. As it can be seen, the latter enabled the visualisation of the orientation of micro-fibres in the muscle bundles, invisible in the former.

 

Fig. 12. Reconstructed axial (a), (d), sagittal (b), (e), and coronal (c), (f) planes of the rat heart for unretrieved (a)-(c) and retrieved (d)-(f) images (scale bar = 2 mm).

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4. Conclusions

The optimization of fixed-length, commercially available, X-ray imaging systems to exhibit superior contrast with weakly-attenuation biological samples is described. The system was optimized for phase-contrast imaging based on single-distance free space propagation; X-ray phase retrieval and CT reconstruction were performed. The variation of image quality with source-to-sample distance (and therefore magnification and propagation distance) was investigated experimentally for a range of focal spot, spectral energy and tube current conditions, supported by previously developed theoretical frameworks and wave optics simulations.

Generally, the optimization of a system with constant R_tot is different from that of systems where R_tot is freely varied; yet most commercial systems are housed in fixed-length cabinets, and from a PBI perspective it makes sense to use the full length of the cabinet.

While for varying R_tot values SNR and contrast could be considered competing parameters, at constant R_tot (as is considered here) the same geometry maximizes SNR and contrast. However, a different geometry optimizes spatial resolution, as made clear by Eqs. (9) and (10) and by the results presented in Fig. 4. Furthermore, the geometry of a fixed R_tot system affects the FoV, which also needs to be taken account in the optimization process. However, once the geometry that optimizes fringe intensity or resolution is identified, larger samples can be brought as close as possible to this as done for the rat heart example of Fig. 12.

A secondary observation emerging from this work was that the importance of the optimization process (in terms of fringe contrast and spatial resolution) relaxes significantly as the focal spot becomes larger, though this reduces the benefit of edge enhancement. Finally, the effect of the geometry to the CNR of the retrieved images was investigated; it was found that small source-to-sample distances, R1 (or equally, high M_geom) are advantageous for increased CNR, in a system with constant R_tot. Ultimately, this work shows that widely available components, typical of several fixed length, cabinet-based commercial systems, can still allow transformative phase-based effects in the imaging of weakly absorbing samples if properly optimized. Our paper also shows that significant contrast gains compared to attenuation-based micro-CT can be achieved (Fig. 12) through continuous scans at relatively high acquisition speed (360 sec), despite the use of a microfocus X-ray source.