A hybrid simulation framework for computer simulation and modelling studies of grating-based x-ray phase-contrast images

Phase-sensitive x-ray imaging has, in recent years, evolved into a promising addition to conventional x-ray imaging. At diagnostic energies, the x-ray phase coefficient of biological materials is a factor 100 to 1000 times higher than the corresponding attenuation coefficient; the additional material information could potentially be used to improve current medical diagnostic imaging. Dedicated phase-contrast techniques have been developed over the past few years and potential candidate imaging systems currently include propagation-based imaging (Snigirev 1995, Cloetens et al 1996), analyser-based imaging (Davis et al 1995, Chapman et al 1997), crystal interferometry (Bonse and Hart 1965, Momose 1995), edge illumination (Olivo and Speller 2007) and grating-based phase-contrast imaging (David et al 2002, Momose et al 2003, Pfeiffer et al 2006). Due to its compatibility with conventional (incoherent, low brilliance) x-ray sources, the latter method is of specific interest.

Grating-based phase-contrast imaging (GB-PCI) yields, via a stepping sequence, a familiar attenuation or transmission (Tr) image and two additional images: a differential phase (dP) and a dark field (DF) image. The former visualizes the refraction effects caused by phase shifts induced by the object, while the latter is related to subpixel structure within the object. Many experimental studies have already demonstrated the clinical potential of this technology and preclinical studies are ongoing. In practice, collecting relevant samples (with pathologies) is often time consuming, expensive and generally only available to research facilities directly associated with a clinical site. Phantom studies may offer an alternative route, provided phantoms with phase and attenuation coefficients matching those of biological samples can be found. Computer simulation studies overcome these limitations by combining (experimental) datasets with simulated lesions. In conventional radiology, this method is exploited frequently (Young et al 2013, Solomon and Samei 2014), however, in GB-PCI, as far as we know, no such studies have yet been performed. A potential hurdle might be that the simulation framework required, has to produce large data sets of realistic images that have large fields of view, all to be done relatively quickly. Simulation frameworks of GB-PCI images described in the literature aim to simulate the imaging setup in high detail. Most frameworks are based on wave-optical numerical analysis (Weitkamp 2004, Malecki, Potdevin and Pfeiffer 2012) while some combine wave simulations with Monte Carlo frameworks in order to include x-ray scattering (Bartl et al 2009, Peter et al 2014). Others use computationally expensive Monte Carlo methods, where refraction is included via ray tracing (Wang et al 2009) or where diffraction effects based on the Huygens-Fresnel principle are included (Cipiccia et al 2014). While these methods offer an accurate and detailed simulation of signal transfer, the complexity of these frameworks make them computationally expensive with limited applicability for simulation of simulation studies of clinical tasks.

This work is based on the hypothesis that full system modelling may not always be required. When many technical details of the system have been established and/or when a prototype system is being produced, the focus may shift to the study of specific (imaging) tasks. The topic of this study is, therefore, not detailed modelling of system components and their influence on image quality, but rather the application of summary data acquired on a prototype GB-PCI system to generate realistic images. To this end, a hybrid image modelling technique is introduced that combines theoretical equations with empirical data to simulate transmission and differential phase images. Dark field images are not included, since the lack of data on the microscale structure of biological tissues responsible for the dark field signals is limited. In this work, three imaging tasks of differing complexities are simulated with a hybrid, fast simulation framework. The results are then compared to experimental data in terms of structural similarity and contrast.

When x-rays propagate through matter, both the phase and amplitude are influenced by the material specific coefficients $\delta (\lambda)$ and $\beta (\lambda)$ respectively, which are related to the complex refractive index $n=1-\delta +{\rm i}\beta$. For a given wavelength, the x-ray wave function ${{\psi }_{0}}$ after transmission through a material with thickness $t~$ is given by $\psi$:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \psi ={{\psi }_{0}}~{{{\rm e}}^{{\rm i}k\delta t-k\beta t}}\nonumber \end{align} \tag{ 1 }$$

in which $k=2\pi /\lambda ~$ is the wavenumber. A changing phase component, perpendicular to the beam propagation direction ($x$), causes the beam to refract under a specific angle $\alpha$ (Bravin 2013):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \alpha =\tan \left( \frac{\partial \delta t}{\partial x} \right).\nonumber \end{align} \tag{ 2 }$$

In the case of x-rays, typical values for $\alpha$ lie in the $\mu$rad range and in order to extract such small effects dedicated methods have been developed.

In GB-PCI, the object to be imaged is placed in a region containing a high frequency intensity pattern. Comparison of the object-disturbed pattern with the original pattern enables the construction of transmission (Tr), differential phase (dP) and dark field (DF) images (David et al 2002, Momose et al 2003, Pfeiffer et al 2006). Three gratings are used to create and measure the intensity pattern (see figure 1). The first grating G₀ creates individually coherent beams, which constructively interfere at the detector plane. The G₁ grating induces a periodic phase shift within the x-ray wave front, creating a high intensity fringe pattern at a fractional Talbot distance $d$. The pitch (${{p}_{2}}$) of the third grating G₂ matches the intensity pattern period and by stepping this grating, the pattern can be sampled and reconstructed. The amplitude to mean intensity of the fringe pattern is the so-called 'visibility' of the system and determines the noise in the dP and DF image. A detailed description of the technology can be found in Pfeiffer et al (2006).

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As the technology relies on wave interference effects, most simulation platforms are based on numerical wave propagation methods. Such methods are flexible and suitable for exploring GB-PCI system geometry and set up, but are demanding in terms of memory usage and simulation time as the high grating pitch, typically between 2 and 5 $~\mu$m, requires high spatial sampling of the wave matrix which is computationally expensive for realistic fields of view. Instead of modelling the wave interactions, it is also possible to compute the expected signal based on theory and this approach forms the basis of our method. For transmission imaging, the expected signal (A) can be predicted from the Beer–Lambert law:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm A}_{{\rm S}}={{{\rm e}}^{-ut}}={{{\rm e}}^{-2k\beta t}}\nonumber \end{align} \tag{ 3 }$$

where $t$ is the thickness of the object. For the differential phase signal ($\varphi$), a similar calculation can be made. The intensity shift at G₂ due to refraction angle $\alpha ~$ is $\tan \left( \alpha \right)\cdot d$, using equation (2):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \varphi_{\rm S}=\frac{2\pi }{{{p}_{2}}}\cdot d\cdot \tan \left( \frac{\partial \delta t}{\partial x} \right)\cdot S\nonumber \end{align} \tag{ 4 }$$

where S is the signal loss due to the offset distance (${{d}_{0}}$) between the object and G₁; $S=1-\frac{{{d}_{0}}}{L}$ (Engelhardt et al 2007, Chabior et al 2012).

2.1. The simulation framework

The hybrid framework developed here combines theoretical equations with experimentally measured data to generate realistic images. First, the phase and attenuation properties of the projected object ($\beta t$ and $\delta t$) are calculated. Each material present in the object is modelled as a grid of thicknesses (${{T}_{i}}(x,y)$) with a maximum object thickness ${{T}_{z}}$. The parameters $x$ and $y$ denote the pixel coordinates. A summation over all material constituents results in a phase matrix ($\Delta$) and an attenuation matrix (${\rm B}$), for example:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \begin{array}{@{}cccccccccccccccccccc@{}} & \Delta \left(x,y \right)=~\underset{i={\rm materials}}{\mathop \sum }\,{{T}_{i}}\left(x,y \right){{\delta }_{i}}~ \nonumber \\ & =~~{{T}_{1}}\left(x,y \right){{\delta }_{1}}+{{T}_{2}}\left(x,y \right){{\delta }_{2}}+\ldots +\left({{T}_{Z}}-{{T}_{1}}\left(x,y \right)-{{T}_{2}}\left(x,y \right)-\ldots \right){{\delta }_{{\rm bg}}}, \nonumber \\ \end{array}\nonumber \end{align} \tag{ 5 }$$

where $bg$ indicates the background material. A similar expression can be derived for ${\rm B}$ using $\beta$ and the same thickness function ${{T}_{i}}$.

Secondly, the finite source size is accounted for by convolving the object with a Gaussian filter ($G(u,v)$) with a standard deviation equal to the scaled focal spot blur. Blurring due to the finite source size and the x-ray detector is included via a 2D presampling modulation transfer function (MTF) obtained from x-ray detector characterization data (see section 2.2.1). A standard Fast Fourier Transform (FFT) technique is used to apply the MTF. In combination with equations (3) and (4):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm A_{{\rm MTF}}}(x,y)={{\mathcal{F}}^{-1}}\left\{\mathcal{F}\left\{{{{\rm e}}^{-2\,{\rm kB}\left(x,y \right)}} \right\}\cdot {\rm MT}{{{\rm F}}_{{\rm Tr}}}(u,v)\cdot ~G(u,v) \right\}\nonumber \end{align} \tag{ 6 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\varphi_{{\rm MTF}}}\left(x,y \right)=~{{\mathcal{F}}^{-1}}\left\{\mathcal{F}\left\{~\frac{2\pi }{{{p}_{2}}}d\tan \left(\frac{\partial \Delta \left(x,y \right)}{\partial x} \right)\cdot S \right\}\cdot {\rm MT}{{{\rm F}}_{{\rm dP}}}\left(u,v \right)\cdot ~G(u,v) \right\}.~\nonumber \end{align} \tag{ 7 }$$

Lastly, noise (${{N}_{i}}$, where $~ i$ indexes the image type) of the correct texture is added, using the approach of Saunders and Samei (2003):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm A}(x,y)={{\mathcal{F}}^{-1}}\{\mathcal{F}\{{{{\rm e}}^{-2\,{\rm kB}\left(x,y \right)}}{{~}^{~}}\}\cdot {\rm MT}{{{\rm F}}_{{\rm Tr}}}(u,v)\cdot ~G(u,v)\}+{{N}_{{\rm Tr }~ }}(x,y)\nonumber \end{align} \tag{ 8 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\varphi }\left(x,y \right)=~{{\mathcal{F}}^{-1}}\left\{\mathcal{F}\left\{~\frac{2\pi }{{{p}_{2}}}d\tan \left(\frac{\partial \Delta \left(x,y \right)}{\partial x} \right)\cdot S \right\}\cdot {\rm MT}{{{\rm F}}_{{\rm dP}}}\left(u,v \right)\cdot ~G(u,v) \right\}+{{N}_{{\rm dP }~ }}(x,y).\nonumber \end{align} \tag{ 9 }$$

In this method, first, a matrix of normally distributed random values ($R$) with zero mean and unit variance is generated and then filtered by the square root of the noise power spectrum (NPS):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle N_{i}^{0}(x,y)=~{{\mathcal{F}}^{-1}}\left\{\mathcal{F}\left(\left\{R\left(x,y \right) \right\}\cdot \sqrt{{\rm NP}{{{\rm S}}_{i}}\left(u,v \right)} \right) \right\}.\nonumber \end{align} \tag{ 10 }$$

The NPS was measured as described in section 2.3.1. We assume that the x-ray detector remains quantum noise limited and therefore the shape of the NPS remains invariant as exposure level at the x-ray detector changes. The correct magnitude of noise in the final noise image (${{N}_{i}}(x,y)$) is obtained by scaling the noise image $N_{i}^{0}$. To ensure a unit variance in $N_{i}^{0}$, the noise image is first scaled with a factor $~1/{{\sigma }_{Ni}}$, with ${{\sigma }_{Ni}}$ the average standard deviation of $N_{i}^{0}.$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{N}_{i}}(x,y)=N_{i}^{0}(x,y)\cdot {{\sigma }_{i}}/{{\sigma }_{Ni}}.\nonumber \end{align} \tag{ 11 }$$

The variance of the noise is calculated for given system parameters and entrance air kerma applied using the method supplied by Revol et al (2010) or by Engel et al (2011). The level of quantum noise (${{\sigma }_{i}}$) depends on the image type and is given by:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sigma _{{\rm Tr}}^{2}\left(x,y \right)=\frac{{{f}_{r}}{\rm A}{{\left(x,y \right)}^{2}}}{{{a}_{0}}{{N}_{{\rm G}2}}}\left(1+\frac{1}{{\rm A}\!\left(x,y \right)} \right)\nonumber \end{align} \tag{ 12 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sigma _{{\rm dP}}^{2}\left(x,y \right)=\frac{2{{f}_{r}}}{{{a}_{0}}{{N}_{{\rm G}2}}~{{v}^{2}}}\left(1+\frac{1}{{\rm A}\!\left(x,y \right)~{{{\rm V}}^{2}}}~ \right)\nonumber \end{align} \tag{ 13 }$$

where A is the transmission signal in each pixel, therefore the flux is modulated depending on attenuation in the object. The dark field induced coherence loss (V) is generally not known and therefore set to one. ${{N}_{{\rm G}2}}$ is the number of steps used for an acquisition, $v$ is the visibility, which was set to the average experimentally measured value in a central region of interest. The coefficient ${{f}_{r}}$ is a detector specific constant that corrects for deviations in the Poisson nature of the detector response (Revol et al 2010). The dose factor ${{a}_{0}}$ is the average pixel value in all projections and can be calculated by ${{a}_{0}}={{a}_{{\rm K}}}\cdot K$, where ${{a}_{{\rm K}}}$ is a calibration factor and $K$ the entrance air kerma applied. The calibration factor ${{a}_{{\rm K}}}$ is the average pixel value generated per unit air kerma and is experimentally determined as described in section 2.2.1.

In experimental setups there will always be some beam divergence, magnifying the object. The magnification of the object is then:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{M}_{O}}=\frac{L+{\rm d}}{L-{{d}_{0}}},\nonumber \end{align} \tag{ 14 }$$

where ${{d}_{0}}$ is some additional offset distance due to the finite object to G₁ distance. This beam divergence is included by a simple approach of rescaling the pixel spacing by a factor $1/{{M}_{0}}$ in the x and y directions. As the flux was scaled to experimental data, this does not influence the total signal in the pixel or the variance of the noise.

2.2. The experimental setup

2.3. Comparison of simulation framework to experimental data

3.1. Analytical object model

For a simple object like the PMMA sphere, close agreement between simulation and experiment is found as demonstrated in figure 2. Quantitatively a SSIM between simulation and experiment of 0.99 and 0.81 is found for Tr and dP respectively. Note that the higher magnitude of noise in the differential phase image compared to the transmission image leads to a reduction in SSIM value.

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3.2. 3D printed heart

In figure 3 the result for the printed heart demonstrates that similar images are produced for more complex shapes. After registration, the simulated data (a,d) matches the experimental data (b,e) with similar contrasts as shown in figure 3.

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An offset in Tr values can be seen in figure 3(c) which we attribute to inaccurate material coefficient modelling—we would expect a reduction in this offset if the exact chemical composition were known. However, it can be seen than similar contrasts are achieved. The maximum contrast in each of the ROIs (separated with blue vertical lines in figures 3(c) and (f), for Tr are respectively 0.21 0.07 0.08 and 0.12 in simulation and in experiment 0.19, 0.06, 0.08 and 0.10. Similarly in dP the maximum contrasts in experiment were respectively 0.34, 0.17, 0.14 and 0.15, while in simulation 0.34, 0.16, 0.13 and 0.12.

3.3. Chest RX mouse

Differential phase and transmission images of a mouse chest using the simulation framework match the appearance of the experimentally acquired datasets (figures 4(a), (b), (d) and (e)). Imperfect registration of the $\mu$CT and GB-PCI data causes some geometric variations, however, importantly the bony, soft tissue and lung field structures within these images are closely matched.

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This is also confirmed quantitatively where the maximum contrast in the three regions indicated in figures 4((c) and (f)) is respectively 0.16 0.33 and 0.21 for the experimental Tr data and 0.14 0.35 and 0.20 for the simulated data. For the dP data similarly the contrast is 0.25 0.20 and 0.27 for the experimental data and 0.17, 0.27 and 0.26 for the simulated data. The contrast is both over- and underestimate, which indicates that this is due to geometrical variations. As the images in the GB-PCI setup are taken horizontally, while in the $\mu$CT vertically, gravity acts differently on the animal and the anatomy in the experimental GB-PCI images is more compressed. It is also apparent that the simulated images have a slightly higher level of blurring, due to the modelling based on the $\mu$CT data with pixel sizes not an inverse multiple of the GB-PCI pixel sizes.

Bone was modelled as 90% rib spongiosa and 10% mineral bone, however, in the experimental data a higher degree of attenuation is seen. The simulated dP images underestimate the level of noise at the level of the lungs, since the dark field coherence loss was not accounted for.

3.4. Computational cost

The proposed framework differs from published frameworks as the fundamental idea is not to produce an extremely detailed simulation at the grating level that can assist in system design and development, but rather use established summary metrics as a means of generating realistic images quickly. This speed-up is paid for with a number of limitations. First and most straightforward, it is not possible to examine hypothetical cases, for example studying what would be the effect of increased G₁–G₂ distance, as the simulation is fixed for the system for which the summary data were measured. The implications on visibility, flux, magnification, etc are therefore not implemented within the simulation and detailed grating modelling using for example a wave propagation framework should be used to first estimate the impact of this change. Simulating dark field images using the hybrid method requires an analytical relationship between the object properties and the measured signal. In Bech et al (2010) the expected DF signal is formulated as a function of the linear diffusion coefficient $\newcommand{\e}{{\rm e}} \epsilon$ of the object and could be used as a first approximation. However, this model yields insufficient dark field accuracy as the expression of Bech et al does not include all sources of visibility loss (Bevins et al 2013, Koenig et al 2016). The focal spot size, pixel size, magnification, beam hardening and strong phase gradients will induce pseudo-dark field contrast. All these effects should be included in order to simulate the dark field images with sufficient accuracy. Moreover, no tabulated values of $\newcommand{\e}{{\rm e}} \epsilon ~$ are currently available in the literature for the relevant materials, further restricting the application of this type of modelling. For this reason, dark field modelling was not included in this work.

One of the consequences of the omission of the dark field component is an underestimation of the noise in regions of the dP images with a high level of small angle scatter. In the case of prior knowledge of the dark field signal (e.g. via previous experiments), this coherence loss of the signal can be included empirically in the noise calculation of the dP signal. The simulation currently assumes a monochromatic energy. Extending the framework to use a polychromatic source is possible but would require knowledge of visibility and ${{a}_{{\rm K}}}$ response versus energy. As this information is difficult to access and the accuracy gained is expected to be limited, this was not considered. As a consequence, mean energy values for $\delta$ and $\beta ~$ were used and beam hardening was not included. Lastly, large angle scatter was not accounted for. Most applications of interest are limited in size and the contribution of scatter to the detector is assumed to be minimal (Vedantham et al 2014, Vignero 2018).

There is a movement towards the use of simulation and modelling to assist in the regulatory evaluation of medical devices in general (Morrison and Dreher 2017) and imaging products in particular (VICTRE, FDA (2018)). These types of studies can be referred to as computational modelling and simulation (CM&S) (Morrison and Dreher 2017) or as virtual clinical studies (Bakic et al 2016, 2018) and are likely to become increasingly important in the development and justification of new medical imaging modalities. In addition to the system characterization required for the modelling (Mackenzie et al 2017) these methods require simulation of the anatomy (Kiarashi et al 2014, Segars et al 2018) and pathology of interest (Shaheen et al 2014). The presented framework allows rapid modelling of dP and Tr images for a well-established GB-PCI setup. After the user has characterized system setup as described in section 2.2.1, images suitable for use in clinical simulation study can be produced quickly, with a CPU time shorter than 3 min to generate 100 images with a 200 by 200 FOV. The framework described here is therefore ideally suited for these types of computational modelling and simulation studies where large numbers of images are needed. For example, nodules can be simulated into experimental or simulated lung images to study the effect of location and size on the detectability performance and to benchmark against the performance of current, standard imaging techniques. This should enable in-depth, quantitative studies into the added benefit of using dP images for specified tissue groups or for other new or even theoretical materials such as new contrast agents. Exploration of specific applications via computational modelling and simulation helps to justify animal and human experiments and can be a crucial step for GB-PCI to evolve into a clinical tool.

Of course there is the question of the required level of accuracy in the simulation, as it is crucial that accuracy is sacrificed for simulation speed. While there are some differences between the experimental and simulated data presented here, the accuracy is believed to be sufficient for these types of studies. In fact the required accuracy is likely to be task dependent and is difficult to generalise. As for any simulation study, realistic modelling of the objects is crucial: here experimentally acquired $\mu$CT data was used for the modelling. The results suffered slightly from the influence of the $\mu$CT resolution and ideally a higher resolution $\mu$CT scanner should be used. Further improvement of the simulated images could also come from manual segmentation of the images.

A simulation framework for producing GB-PCI differential phase and transmission images has been introduced with the goal of generating images for use in computational modelling and simulation studies. The framework uses 3D voxel models and typical metrics measured/estimated for the system in question, namely the detector presampling MTF, NPS, visibility and a dose-to-noise calibration factor. The major advantage of the proposed simulation framework is the gain in simulation speed and reduced memory requirements. With only a small loss of accuracy, transmission and differential phase images with large fields of view can be generated quickly without the requirement of high spatial sampling.

This work has been supported by MaXIMA project: 3D breast cancer models for x-ray imaging research. This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 692097.