## Highlights

- •Drawback of conventional X-ray radiography overcome by using X-ray phase contrast.
- •Effective for materials consisting of low-Z element, such as polymers and biological soft tissues.
- •Quantitative images mapping refraction and scattering in addition to attenuation.
- •Development initially at synchrotron radiation facilities expanding at hospitals.
- •Apparatus using grating interferometry in operation for clinical studies with patients in hospitals.

## Abstract

## Keywords

## 1. Introduction

## 2. What is X-ray phase imaging?

### 2.1 Significance of using X-ray phase information

*n*(=1 – δ +

*i β*), where imaginary part

*β*describes the X-ray attenuation and the phase shift is given by δ. Both δ and

*β*are small quantities in the hard X-ray region, reflecting the fact that X-rays penetrate inside objects nearly straight. What is important here is another fact that the ratio δ/β is around 1,000 for materials consisting of light elements, such as polymers and biological soft tissues.

*T*. X-ray attenuation follows Lambert-Beer’s law, and optical density

*D*:

is measured in conventional X-ray radiography, where

*I*

_{0}and

*I*are the X-ray intensities in front of and behind the object and μ is the linear absorption coefficient. Strictly speaking, the linear absorption coefficient is a function of position μ (

*x*,

*y*,

*z*) and Eq. (1) is rewritten as

where it is assumed that X-rays penetrate in the

*z*direction. By using

*β*and X-ray wavelength λ,

*n*= 1), phase shift Φ (

*x*,

*y*) is given by

where

*Nk*,

*Zk*,

*f’k*, and μ

^{a}

*k*are the atomic density, atomic number, real part of the atomic scattering factor, and the atomic absorption coefficient of element

*k*existing in the object. μ

^{a}

*k*has the dimension of area and corresponds to the interaction cross section. Note that the interactions relating to X-ray attenuation are the photoelectron effect, Thomson scattering, and Compton scattering, and μ

^{a}

*k*is the quantity defined by summing the effects of the three interactions. In Eq. (5),

*pk*($\equiv {r}_{e}\lambda \left({Z}_{k}+{f\text{'}}_{k}\right)$) is used as the interaction cross section of the phase shift [

*D*and Φ is derived from the difference between μ

^{a}

*k*and

*pk*. Note that the constitution and density affects both

*D*and Φ in the same manner.

^{a}

*k*and

*pk*for X-ray photon energies of 20 keV, 40 keV, and 60 keV as functions of the atomic number. The magnitudes of

*pk*are larger than those of μ

^{a}

*k*by about 1,000 times for low-Z elements. This fact suggests that the use of X-ray phase information (phase contrast) has the potential to overcome the drawback of the poor sensitivity of conventional X-ray radiography based on the absorption contrast especially for materials consisting of low-Z elements. Table 1 lists the values of δ and

*β*calculated for some materials at 30 keV and 60 keV photon energies [

*β*exceeds 1,000 for low-Z materials, inheriting the difference shown in Fig. 1.

*i β*) of some materials at 30 keV and 60 keV photon energies.

Material | 30 keV | 60 keV | ||||
---|---|---|---|---|---|---|

δ | β | δ/β | δ | β | δ/β | |

Water | 2.56 × 10^{−7} | 1.06 × 10^{−10} | 2.41 × 10^{3} | 6.42 × 10^{−8} | 2.93 × 10^{−11} | 2.19 × 10^{3} |

Polyimide | 3.38 × 10^{−7} | 1.15 × 10^{−10} | 2.34 × 10^{3} | 9.70 × 10^{−8} | 5.01 × 10^{−11} | 1.94 × 10^{3} |

Polypropylene | 2.37 × 10^{−7} | 7.01 × 10^{−11} | 3.38 × 10^{3} | 5.95 × 10^{−8} | 2.39 × 10^{−11} | 2.48 × 10^{3} |

Teflon | 4.87 × 10^{−7} | 2.46 × 10^{−10} | 1.98 × 10^{3} | 1.22 × 10^{−7} | 7.09 × 10^{−11} | 1.72 × 10^{3} |

Ti | 9.68 × 10^{−7} | 7.08 × 10^{−9} | 1.37 × 10^{2} | 2.41 × 10^{−7} | 5.60 × 10^{−10} | 4.30 × 10^{2} |

Fe | 1.70 × 10^{−6} | 2.01 × 10^{−8} | 8.46 × 10^{1} | 4.22 × 10^{−7} | 1.54 × 10^{−9} | 2.74 × 10^{−2} |

*x*,

*y*) is equivalent to the change in the form of the wavefront. Considering the fact that a wave propagates in the direction perpendicular to its wavefront, the slope of the wavefront induced by the phase shift implies a change in propagation direction φ(

*x*,

*y*) (i.e., ‘refraction’), which is given by

where ${\mathrm{\nabla}}_{\perp}$ is the gradient operator in the

*x*-

*y*plane. The value of φ, which strictly depends on the shape of the object, is in the order of microradians or less. In conventional X-ray radiography, therefore, it is reasonably assumed that X-rays penetrate straight in objects. In X-ray phase-contrast techniques, however, this slight change in propagation direction is used as a signal.

### 2.2 X-ray phase-imaging techniques

#### 2.2.1 Method using two-beam interference

*I*(

*x*,

*y*) observed by this interferometer is written as

where

*A*(

*x*,

*y*) and

*B*(

*x*,

*y*) are the average intensity and fringe amplitude, respectively. It is not technically easy to fabricate an ideal interferometer without deformation and a built-in moiré pattern is normally observed, as shown in the image of a polymer sphere (Fig. 2(a)), even in the absence of an object. Such an effect is expressed by Δ(

*x*,

*y*) in Eq. (8).

*I*(

*x*,

*y*) is not sufficient in performing X-ray phase imaging. Objects normally cause both attenuation and phase shift. The former appears in

*A*(

*x*,

*y*), and it is not easy to extract the effect of the latter (Φ (

*x*,

*y*)) purely. Here, two approaches for differentiating between attenuation and phase shift are explained, that is, the fringe-scanning method [

#### 2.2.1.1 Fringe-scanning method

where the stepwise change in the phase difference is denoted by

*k*, and

*M*is an integer equal to or larger than 3. Then, by acquiring multiple interference patterns described by Eq. (9), Φ (

*x*,

*y*) can be extracted by

where arg[ ] implies the extraction of the argument. Note that Δ(

*x*,

*y*) can be determined without an object (that is, Φ (

*x*,

*y*) = 0) using the same procedure. Thanks to this phase-imaging procedure, we can avoid the difficulty of preparing a perfect interferometer. On the other hand, an absorption image is obtained by averaging

*Ik*(

*x*,

*y*) (

*k*= 1, 2, ···,

*M*).

#### 2.2.1.2 Fourier-transform method

*f*

_{0}are generated in the

*x*direction, the interference pattern is written as

where

and the asterisk denotes the complex conjugate. The one-dimensional Fourier transform of Eq. (11)’ yields

where subscript F indicates the Fourier transform of each term in Eq. (11)’. The three terms on the right-hand side of Eq. (13) are separated by

*f*

_{0}from each other along frequency axis

*f*. If the sizes of the structures involved in the object are sufficiently larger than 1/

*f*

_{0}, the overlap of the three terms on the

*f*axis is ignored, and

*C*

_{F}(

*f*-

*f*

_{0},

*y*) can be extracted. Next,

*C*

_{F}(

*f*,

*y*) is obtained by shifting

*C*

_{F}(

*f*-

*f*

_{0},

*y*) by

*f*

_{0}to the origin. The inverse Fourier transform of

*C*

_{F}(

*f*,

*y*) with respect to

*f*yields

*C*(

*x*,

*y*). Then, Φ (

*x*,

*y*) + Δ(

*x*,

*y*) is obtained by calculating the argument of

*C*(

*x*,

*y*). The determination of Δ(

*x*,

*y*) is feasible without an object. An absorption image is obtained by the inverse Fourier transform of

*A*

_{F}(

*f*,

*y*).

*x*,

*y*) is limited by 1/

*f*

_{0}while the fringe-scanning method can provide phase images with much better spatial resolutions without degradation from interference patterns. Depending on the purpose of phase imaging, we need to select one of the two approaches.

*N*is the total number of X-ray photons measured in a detector pixel.

^{−1}in practice, does not provide the phase shift directly but the phase shift wrapped within (-π, π). Therefore, a phase-unwrapping procedure [

#### 2.2.2 Method using crystal diffraction enhancement

*R*(θ - θ

_{B}), which is given by dynamical diffraction theory [

_{B}, the reflectivity reaches its maximum and decreases rapidly with increasing | θ - θ

_{B}|. When an object is placed in front of the crystal,

*I*

_{D}(

*x*,

*y*; θ - θ

_{B}) ≡

*I*(

*x*,

*y*)·

*R*(θ - θ

_{B}- φ

*x*(

*x*,

*y*)) is measured, where

*I*(

*x*,

*y*) is the X-ray intensity just behind the object and φ

*x*(

*x*,

*y*) is the

*x*component of $\frac{\lambda}{2\pi}{\mathrm{\nabla}}_{\perp}\mathrm{\Phi}(x,y)$ (see Eq. (7)). Here, it is assumed that X-ray diffraction occurs in the plane parallel to the

*x*axis and perpendicular to the

*y*axis.

*m*across the Bragg diffraction condition, φ

*x*(

*x*,

*y*) can be determined by

*I*

_{D}(

*x*,

*y*; θ

*m*), provided that the angular scan is performed in a fully wide range. More simply, when two images

*I*

_{L}(

*x*,

*y*) and

*I*

_{R}(

*x*,

*y*) are measured at angular positions θ

_{L}and θ

_{H}at the slopes on the low-angle and high-angle sides of the profile of

*R*, φ

*x*(

*x*,

*y*) can be determined approximately by [

#### 2.2.3 Propagation-based method

*β*. Then, the TIE has the form:

and, with a single-distance measurement, the phase shift just behind an object can be calculated approximately by

where $\mathcal{F}$ and ${\mathcal{F}}^{-1}$ denote the forward and backward Fourier transforms and

*u*and

*v*are the Fourier conjugate coordinates of

*x*and

*y*, respectively [

#### 2.2.4 Grating-based method

*I*

^{G}(

*x*,

*y*) measured by an X-ray Talbot interferometer, for which it is assumed that linear gratings in the

*y*direction are employed, is given by

where ${B}_{n}^{\text{G}}(x,y)$ are calculated from the products of the Fourier coefficients of the self-image of G1 and the intensity transmission function of G2, and φ

*x*(

*x*,

*y*) is the

*x*component of φ (

*x*,

*y*) given in Eq. (7).

*d*is the period of G2, and χ is the displacement of one of the gratings against the other in the

*x*direction. Δ(

*x*,

*y*) is introduced in Eq. (19), as it is in Eq. (8), to express the imperfection in grating fabrication and alignment.

*x*(

*x*,

*y*). By displacing G2 with a step of

*d*/

*M*, multiple moiré patterns are recorded. In Eq. (19), χ is replaced with

*kd*/

*M*, and the same procedure as Eq. (10) is applied [

where ${I}_{k}^{\mathrm{G}}\left(x,y\right)$ is the image when χ =

*kd*/

*M*. By displacing G1 instead of G2, φ

*x*(

*x*,

*y*) can be determined in the same manner. More details on the image mapping refraction in addition to those mapping absorption and scattering are described later in Section 2.4.

- Berujon S.
- Wang H.
- Sawhney K.

*Phys. Rev. A.*2012; 86https://doi.org/10.1103/PhysRevA.86.063813

### 2.3 X-ray phase tomography

*D*along multiple projection lines across an object, linear absorption coefficient μ (

*x*,

*y*,

*z*) can be reconstructed. The same form as Eq. (2) is found for the phase shift in Eq. (4), which implies that δ(

*x*,

*y*,

*z*) can be reconstructed by measuring Φ along the projection lines across an object using the same mathematical procedure as that for conventional X-ray CT [

- Momose A.

*Nucl Instrum Methods Phys Res, Sect A.*1995; 352: 622-628https://doi.org/10.1016/0168-9002(95)90017-9

*x*) is measured. In these cases, by using a Hilbert kernel in the filtered back-projection method [

*x*,

*y*,

*z*) can be reconstructed as well.

- Shinohara M.
- Yamashita T.
- Tawa H.
- Takeda M.
- Sasaki N.
- Takaya T.
- Toh R.
- Takeuchi A.
- Ohigashi T.
- Shinohara K.
- Kawashima S.
- Yokoyama M.
- Hirata K.-I.
- Momose A.

*American Journal of Physiology-Heart and Circulatory Physiology.*2008; 294: H1094-H1100https://doi.org/10.1152/ajpheart.01149.2007

*f’*in Eq. (5) is negligible for low-Z elements in the hard X-ray region, δ is approximately proportional to the electron density, which is furthermore approximated to the mass density. According to the results by using synchrotron radiation and a Bonse-Hart interferometer (Fig. 2(a)), a density deviation larger than 1 mg/cm

^{3}, which was evaluated by the standard deviation of noise, could be depicted [

### 2.4 Scattering signal retrieval

*x*(

*x*,

*y*). The meanings of ${B}_{0}^{\mathrm{G}}(x,y)$ and φ

*x*(

*x*,

*y*) were discussed above; that is, information of absorption and refraction (or differential phase) are involved, respectively. Here, the significance and usage of ${B}_{1}^{\text{G}}(x,y)$ are considered.

*V*(

*x*,

*y*) ≡ {

*I*

_{max}(

*x*,

*y*) –

*I*

_{min}(

*x*,

*y*)}/{

*I*

_{max}(

*x*,

*y*) +

*I*

_{min}(

*x*,

*y*)} = ${B}_{1}^{\text{G}}(x,y)/{B}_{0}^{\mathrm{G}}(x,y)$ is calculated, where

*I*

_{max}(

*x*,

*y*) and

*I*

_{min}(

*x*,

*y*) are the maximum and minimum values in the phase-stepping curve. This value is comparable to the visibility of moiré fringes. Specifically, to obtain

*V*(

*x*,

*y*) quickly,

*k*th step.

*x*(

*x*,

*y*). Scattered X-rays no longer contribute to interference and ${B}_{1}^{\text{G}}(x,y)$ decreases as a result. Therefore,

*V*(

*x*,

*y*) is considered to be a map of the distribution of the microstructures. In other words, this signal is called the ‘dark-field’ signal [

*V*(

*x*,

*y*). A similar signal can also be obtained by the approaches shown in Fig. 2(a) and 2(b) [

*V*

_{0}is the visibility without an object and α is introduced as the linear diffusion coefficient to describe its physical sense [

*x*direction in the abovementioned formalization), the dark-field signal varies when an object is rotated about the optical axis; that is, assuming that the variation is sinusoidal,

where ω is the rotation angle about the optical axis. $\mathrm{\Theta}\left(x,y\right)$ describes the anisotropy of the microstructures to be measured. By setting ω = π

*k*/

*M*(

*k*= 1, 2, ···,

*M*),

is used for the analysis using the fringe-scanning method.

*V*

_{0}(

*x*,

*y*) is the average visibility and

*V*

_{1}(

*x*,

*y*) indicates the degree of anisotropy. This treatment is known as ‘vector radiography’ [

## 3. Developments for clinical uses

### 3.1 Mammography

### 3.2 Diagnosis of arthritis

*x*(

*x*,

*y*)) to anatomical findings [

- Yoshioka H.
- Kadono Y.
- Kim Y.T.
- Oda H.
- Maruyama T.
- Akiyama Y.
- Mimura T.
- Tanaka J.
- Niitsu M.
- Hoshino Y.
- Kiyohara J.
- Nishino S.
- Makifuchi C.
- Takahashi A.
- Shinden Y.
- Matsusaka N.
- Kido K.
- Momose A.

*Sci Rep.*2020; 10https://doi.org/10.1038/s41598-020-63155-9

### 3.3 Diagnosis of lung diseases

*in vivo*mouse imaging, it was found that lungs produce significant contrast in dark-field images [

- Bech M.
- Tapfer A.
- Velroyen A.
- Yaroshenko A.
- Pauwels B.
- Hostens J.
- Bruyndonckx P.
- Sasov A.
- Pfeiffer F.

*Sci Rep.*2013; 3https://doi.org/10.1038/srep03209

- Schleede S.
- Meinel F.G.
- Bech M.
- Herzen J.
- Achterhold K.
- Potdevin G.
- Malecki A.
- Adam-Neumair S.
- Thieme S.F.
- Bamberg F.
- Nikolaou K.
- Bohla A.
- Yildirim A.O.
- Loewen R.
- Gifford M.
- Ruth R.
- Eickelberg O.
- Reiser M.
- Pfeiffer F.

- Yaroshenko A.
- Hellbach K.
- Yildirim A.Ö.
- Conlon T.M.
- Fernandez I.E.
- Bech M.
- Velroyen A.
- Meinel F.G.
- Auweter S.
- Reiser M.
- Eickelberg O.
- Pfeiffer F.

*Sci Rep.*2015; 5https://doi.org/10.1038/srep17492

- Bachche S.
- Nonoguchi M.
- Kato K.
- Kageyama M.
- Koike T.
- Kuribayashi M.
- Momose A.

*Sci Rep.*2017; 7https://doi.org/10.1038/s41598-017-07032-y

- Gromann L.B.
- De Marco F.
- Willer K.
- Noël P.B.
- Scherer K.
- Renger B.
- Gleich B.
- Achterhold K.
- Fingerle A.A.
- Muenzel D.
- Auweter S.
- Hellbach K.
- Reiser M.
- Baehr A.
- Dmochewitz M.
- Schroeter T.J.
- Koch F.J.
- Meyer P.
- Kunka D.
- Mohr J.
- Yaroshenko A.
- Maack H.-I.
- Pralow T.
- van der Heijden H.
- Proksa R.
- Koehler T.
- Wieberneit N.
- Rindt K.
- Rummeny E.J.
- Pfeiffer F.
- Herzen J.

*Sci Rep.*2017; 7https://doi.org/10.1038/s41598-017-05101-w

Willer K, Fingerle AA, Gromann LB, De Marco F, Herzen J, Achterhold K, Gleich B, Muenzel D, Scherer K, Renz M, Renger B, Kopp F, Kriner F, Fischer F, Braun C, Auweter S, Hellbach K, Reiser MF, Schroeter T, Mohr J, Yaroshenko A, Maack HI, Pralow T, van der Heijden H, Proksa R, Koehler T, Wieberneit N, Rindt K, Rummeny EJ, Pfeiffer F, Noël PB. X-ray dark-field imaging of the human lung—A feasibility study on a deceased body. PLoS One 2018;13:e0204565.

- Fingerle A.A.
- De Marco F.
- Andrejewski J.
- Willer K.
- Gromann L.B.
- Noichl W.
- Kriner F.
- Fischer F.
- Braun C.
- Maack H.-I.
- Pralow T.
- Koehler T.
- Noël P.B.
- Meurer F.
- Deniffel D.
- Sauter A.P.
- Haller B.
- Pfeiffer D.
- Rummeny E.J.
- Herzen J.
- Pfeiffer F.

*Eur Radiol Exp.*2019; 3https://doi.org/10.1186/s41747-019-0104-7

- Gromann L.B.
- De Marco F.
- Willer K.
- Noël P.B.
- Scherer K.
- Renger B.
- Gleich B.
- Achterhold K.
- Fingerle A.A.
- Muenzel D.
- Auweter S.
- Hellbach K.
- Reiser M.
- Baehr A.
- Dmochewitz M.
- Schroeter T.J.
- Koch F.J.
- Meyer P.
- Kunka D.
- Mohr J.
- Yaroshenko A.
- Maack H.-I.
- Pralow T.
- van der Heijden H.
- Proksa R.
- Koehler T.
- Wieberneit N.
- Rindt K.
- Rummeny E.J.
- Pfeiffer F.
- Herzen J.

*Sci Rep.*2017; 7https://doi.org/10.1038/s41598-017-05101-w

Willer K, Fingerle AA, Gromann LB, De Marco F, Herzen J, Achterhold K, Gleich B, Muenzel D, Scherer K, Renz M, Renger B, Kopp F, Kriner F, Fischer F, Braun C, Auweter S, Hellbach K, Reiser MF, Schroeter T, Mohr J, Yaroshenko A, Maack HI, Pralow T, van der Heijden H, Proksa R, Koehler T, Wieberneit N, Rindt K, Rummeny EJ, Pfeiffer F, Noël PB. X-ray dark-field imaging of the human lung—A feasibility study on a deceased body. PLoS One 2018;13:e0204565.

## 4. Future prospects

*,*toward a tomographic scheme), and so on. Then, other application targets with X-ray phase imaging can be proposed.

- Wang Z.
- Hauser N.
- Singer G.
- Trippel M.
- Kubik-Huch R.A.
- Schneider C.W.
- Stampanoni M.

*Nat Commun.*2014; 5https://doi.org/10.1038/ncomms4797

## Declaration of Competing Interest

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