X-ray phase-shifts-based method of volumetric breast density measurement

Abstract

Purpose: The high breast density is one of the biggest risk factors for breast cancer. Identifying patient having persistent high breast density is important for breast cancer screening and prevention. In this work the authors propose for the first time an x-ray phase-shifts-based method of breast density measurement.

Methods: When x ray traverses the breast, x ray gets not only its intensity attenuated but also its phase shifted. Studying the x-ray phase-shifts generated by the breast tissues, we derived a general formula for determining the volumetric breast density from the breast phase map. The volumetric breast density is reconstructed by retrieving the breast phase map from just a single phase-sensitive projection of the breast, through the use of an innovative phase retrieval method based on the phase-attenuation duality. In order to numerically validate this phase-shifts-based method for measuring the volumetric breast density, the authors performed computer simulations with a digitally simulated anthropomorphic breast phantom.

Results: Using the proposed phase-shifts-based method, we reconstructed the breast phantom's volumetric breast density, which differs from the phantom's intrinsic breast density by only 0.06%. In the presence of noises in the projection image, the reconstructed volumetric breast density differs from the phantom's intrinsic breast density by only 1.79% for a projection signal-to-noise-ratio (SNR) of 34. The error in reconstructed breast density is further reduced to 1.61% and 1.55% for SNR = 68 and SNR = 134, respectively, achieving good accuracies in the breast density determination.

Conclusions: The authors proposed an x-ray phase-shifts-based method of measuring the volumetric breast density. The simulation results numerically validated the proposed method as a novel method of breast density measurement with good accuracies.

INTRODUCTION

As is well known, breast cancer is the most common cancer among American women and is the leading cause of cancer related death among women between 35 and 50 yr of age in the United States. In recent years scientific studies have found that the breast cancer risk is directly linked to the breast density, the relative amount of the dense fibroglandular tissue versus fatty adipose tissue, namely, the fraction of the volume of fibroglandular tissue in the breast. As the fibroglandular tissues appear “hyperdense” in mammograms because of its relatively high attenuation coefficients, studies found that high breast density inferred from mammograms is associated with twofold to fourfold increase in risk for breast cancer.^{1, 2, 3} Although the magnitude of that risk is still under some debate, the scientific literature holds that high breast density is one of the biggest risk factors for breast cancer.² Identifying patient having persistent breast density may benefit these patients from a more frequent breast cancer screening for early detection of breast cancer or from some other preventive measure such as the chemoprevention. The public awareness of breast density as a high risk factor for breast cancer has recently been pushing new legislature and regulatory efforts for providing patients the breast density information from their breast imaging exams. Therefore, in recent years there is strong interest in developing methods to measure volumetric breast density (VBD). The most straightforward method of measuring volumetric breast density would be from the three-dimensional x-ray breast image techniques such as breast computed tomography (CT), which is still in its early period of development. Currently, it is most practical to determine volumetric breast density from the two-dimensional imaging techniques such as mammography.^{4, 5, 6, 7, 8, 9, 10} The mammography-associated methods are all based on the small differences in tissue attenuation of the dense breast fibroglandular tissue and fatty adipose tissue. In some methods, breast density is measured by the categorical scores based on the tissue radiographic opacity on a mammogram acquired with a screen/film based or digital system.^{4, 9, 10} However, such methods are limited by lack of consensual reference standards and are often flawed by the subjectivity. The improved methods reported in the literature all need to conduct elaborated phantom-based system calibrations for breast density measurement. Some of these methods require simultaneous exposure of a calibration phantom alongside the breast for the system calibrations.^{7, 8, 10} These laborious calibrations are indeed required, because mammography exams are conducted with different tube targets/filters and voltages, and the linear attenuation coefficients of breast tissues vary in complicated ways as the x-ray energy changes. Therefore, in search for a more effective method of breast density measurement, it would be desirable to develop a method to measure breast density directly rather than measuring tissue attenuation as a surrogate of the breast density.

In this work we propose to determine the breast density by direct probing the projected electron densities of breast fibroglandular tissue and adipose tissue through the use of the retrieved phase map of a breast. Note that the electron density of a tissue is the intrinsic property of the tissue and it is independent of x-ray photon energies employed. In Secs. 2, 3, 4, we lay down a framework of a phase-shifts-based method of determining the volumetric breast density. We show how to retrieve the breast phase map from a single phase-sensitive projection of the breast. In order to validate this novel breast density measurement method, we discuss the volumetric breast density values reconstructed from computer simulations with a digitally simulated anthropomorphic breast phantom.

METHODS

Our method of breast density measurement is based on an x-ray phase-sensitive imaging technique. As x rays traverse the breast, x rays undergo phase-shifts in addition to the intensity attenuation. The amount of x-ray phase-shift φ generated by tissues is equal to φ = −⟨λ⟩r_e∫_rayρ_e(s)ds = −⟨λ⟩r_eρ_{e, p}, where r_e = 2.818 × 10⁻¹⁵ m is the classical electron radius and λ denotes the x-ray wavelength.^{11, 12, 13, 14, 15} In this paper the angle bracket ⟨·⟩ denotes the spectral average over the detected polychromatic x-ray spectrum, so ⟨λ⟩ denotes the average x-ray wavelength. Here, ρ_e denotes the tissue electron density and ρ_{e, p} = ∫_rayρ_e(s)ds is the projected electron density summed over the x-ray path. Therefore, once we have retrieved the phase map $\phi \left(\vec{r}\right)$ of a breast through its projection image, where $\vec{r}$ denotes the position on the phase map, we are able to find the projected electron density map ${\rho }_{e,p}\left(\vec{r}\right)$ of the breast as ${\rho }_{e,p}\left(\vec{r}\right)=\left(-1/⟨\lambda ⟩r_e\right)\phi \left(\vec{r}\right)$. Using the ${\rho }_{e,p}\left(\vec{r}\right)$-map of a breast, we can determine the volumetric breast density of the breast as follows. For the purpose of breast density determination, we may assume that there are only two types of tissues in breast: the fibroglandular tissue with an electron density ρ_e,fg and the adipose tissue with an electron density ρ_e,ad. Note that the values of ρ_e,fg and ρ_e,ad are available from the measured data in the literature, ρ_e,fg = 3.448 × 10²³/cm³ and ρ_e,ad = 3.108 × 10²³/cm³.¹⁶ Let $m\left(\vec{r}\right)$ be the fibroglandular-tissue fraction along a projection-ray detected at position $\vec{r}$, then we have

$${\rho }_{e,p}\left(\vec{r}\right)=\left[m\left(\vec{r}\right)·{\rho }_{e,\mathrm{fg}}+\left(1-m\left(\vec{r}\right)\right)·{\rho }_{e,\mathrm{ad}}\right]·T\left(\vec{r}\right),$$ (1)

where $T\left(\vec{r}\right)$ denotes the ray path-length in the breast. In mammography, the breast is always kept forcefully and uniformly compressed for spreading breast tissues and reducing the scatters. The compressed breast thickness is always automatically measured and indicated by the imaging system. As long as the source-breast distance is significantly larger than the breast lateral size, the values of $T\left(\vec{r}\right)$ are approximately equal to the compressed thickness T_c of the breast, except for the breast anterior edge, which is not in contact with the compression paddle and constitutes only a very small portion of the breast volume. With this approximation we found that the fibroglandular-tissue fraction along the ray-$\vec{r}$ as given by

$$m\left(\vec{r}\right)=\frac{-\phi \left(\vec{r}\right)/⟨\lambda ⟩r_e-{\rho }_{e,\mathrm{ad}}{T}_c}{\left({\rho }_{e,\mathrm{fg}}-{\rho }_{e,\mathrm{ad}}\right)T_c}.$$ (2)

While $m\left(\vec{r}\right)$ specifies the fibroglandular-tissue fraction along the ray-$\vec{r}$, the breast density, as a risk factor defined in the literature, should be quantified the fibroglandular-tissue fractions averaged over the whole breast. Hence, a map of these $m\left(\vec{r}\right)$-values, which we call the $m\left(\vec{r}\right)$-image of a breast, is the map of the breast's fibroglandular-tissue fractions. The VBD can then be computed from the $m\left(\vec{r}\right)$-image as $\mathrm{VBD}={\sum }_{\mathrm{breast}}m\left(\vec{r}\right)/N_p$, where the sum is taken over the projected breast image and N_p is the total pixel number in the projected breast. Hence, the volumetric breast density can be found from the breast phase map as

$$\mathrm{VBD}=\frac{{\sum }_{\mathrm{breast}}\left(-\phi \left(\vec{r}\right)/⟨\lambda ⟩r_e-{\rho }_{e,\mathrm{ad}}{T}_c\right)}{N_p{T}_c\left({\rho }_{e,\mathrm{fg}}-{\rho }_{e,\mathrm{ad}}\right)}.$$ (3)

In order to implement a phase-shifts-based method of measuring the breast density, it calls for a low radiation dose yet phase-sensitive technique to recover the phase-shifts generated by the breast, as the average glandular doses involved should be closely controlled and monitored as is in current mammography. Although the differences in x-ray phase-shifts between different tissues are about 1000 times greater than their differences in the attenuation, current technology is unable to directly measure the phase-shifts because of the extremely high frequencies of x rays.^{11, 12, 13} Breast phase-shifts have to be retrieved from the phase-sensitive images of the breast. The proposed Eq. 3-based method of breast density measurement can be applied to any of phase-sensitive imaging techniques such as the inline or the grating-based techniques. Among these phase-sensitive techniques, the inline phase-sensitive imaging is the simplest to be implemented, since the inline technique does not need any x-ray optics such as the crystal analyzers or high line-density transmission gratings. In this work, we study the inline phase-sensitive imaging only. The setting for the inline technique is similar to that of conventional mammography, provided one uses a small focal spot and a sufficiently large object-detector distance.^{12, 14, 15} In the inline phase-sensitive imaging, as x rays traverse the breast, x rays undergo phase-shifts and intensity-attenuation, and then diffract freely over a distance upon detection. The resulted image exhibits not only the tissue attenuation contrast, but also the tissue phase contrast, which manifests as the dark-bright fringes at tissues’ boundaries and interfaces in the image.^{11, 12, 13, 14, 15} In order to measure the breast density by using the phase map of a breast, one needs to retrieve the breast phase map from its phase-sensitive projection images.

The phase retrieval is based on x-ray propagation equation, which reveals how the phase-shifts are encoded in the image intensity variations. A convenient form of the propagation equation is the so-called transport of intensity equation (TIE):^{17, 18}

$$\begin{array}{ccc}\hfill I\left({\vec{r}}_D\right)& =& \frac{I_{\mathrm{in}}}{M^2}\left\{A_o^2\left({\vec{r}}_D/M\right)\right.\hfill \\ & & \left.-\frac{R_2⟨\lambda ⟩}{2\pi M}\nabla ·\left(A_o^2\left({\vec{r}}_D/M\right)\nabla \phi \left({\vec{r}}_D/M\right)\right)\right\},\hfill \end{array}$$ (4)

where $I\left({\vec{r}}_D\right)$ is the detected x-ray intensity, I_in is the entrance x-ray intensity, $A_o^2\left(\vec{r}\right)$ is the attenuation image of the imaged object, say a breast, and $\phi \left(\vec{r}\right)$ is the x-ray phase-shift map of the breast. In Eq. 4R₂ is the object-detector distance, M is the magnification factor employed in the projection, and ${\vec{r}}_D$ denotes the position in the detector plane. The operator ∇ denotes the two-dimensional transverse gradient differential operator. From Eq. 4 it is clear that the detected x-ray intensity $I\left({\vec{r}}_D\right)$ is determined not only by the conventional attenuation-based contrast $A_o^2\left(\vec{r}\right)$ but also by the transverse Laplacian and gradient differentials of the phase-shift $\phi \left(\vec{r}\right)$. These differentials represent the edge-enhancements exhibit in the phase-sensitive projection images. Since the phase contrast and attenuation contrast are mixed together in a phase-sensitive projection, there are two unknown variables for a given projection intensity in Eq. 4. Hence, the TIE-based phase retrieval method requires multiple projections (at least two projections) acquired with varying object-detector distances for retrieving the phase-shift map $\phi \left(\vec{r}\right)$ of a breast.^{17, 18, 19} This requirement of multiple image-acquisitions for the TIE-based phase retrievals is obviously cumbersome in implementation, and the multiple exposures make the radiation doses to breast multiplied. As if these were not enough challenges, the TIE-based phase retrieval method is indeed unstable against the noise in projection images. While one might want to increase the radiation doses employed in the projections for retrieving a quality phase map of a breast, the glandular dose to a breast should be kept as low as reasonably achievable because the breast is a radiation sensitive organ. Considering these disadvantages with the TIE-based phase retrieval method, we have to employ a more effective phase retrieval method of measuring breast density through the use of Eqs. 2, 3.

We noted that the breast fibroglandular tissues and adipose tissues are consisting of dominantly elements with Z < 10. If we employ high energy x rays of 60 keV or higher for imaging, the x-ray-breast tissue interactions are dominated by the x-ray Compton scattering from freelike atomic electrons, while the x-ray photoelectric absorption and coherent scattering are all diminished and negligible.²⁰ In this situation, both the tissue attenuation and phase-shift are all determined by tissues’ electron density distributions such that

$$\begin{array}{ccc}& & A^2\left(\vec{r}\right)=exp\left[-{\sigma }_{\mathrm{KN}}\left(E\right){\rho }_{e,p}\left(\vec{r}\right)\right],\hfill \\ & & \phi \left(\vec{r}\right)=-\lambda r_e{\rho }_{e,p}\left(\vec{r}\right),\hfill \end{array}$$ (5)

where σ_KN(E) denotes the Klein–Nishina total cross section for Compton scattering and E is the photon energy The Klein-Nishina total cross section changes slowly with photon energy as²⁰

$$\begin{array}{ccc}\hfill {\sigma }_{\mathrm{KN}}\left(E\right)& =& 2\pi r_e^2\left\{\frac{1+\eta }{{\eta }^2}\left[\frac{2(1+\eta )}{1+2\eta }-\frac{1}{\eta }\log (1+2\eta )\right]\right.\hfill \\ & & \left.+\frac{1}{2\eta }\log (1+2\eta )-\frac{1+3\eta }{{(1+2\eta )}^2}\right\},\hfill \end{array}$$ (6)

where η ≡ E/511 keV. In a recent work we explored the relationship between the phase-shift and attenuation defined by Eq. 5 and called this as the phase-attenuation duality (PAD).²¹ We found that when the phase-attenuation duality holds, the x-ray propagation equation gets simplified and the phase map can be retrieved from just a single phase-sensitive projection $I\left({\vec{r}}_D\right)$ of the imaged object:^{21, 22}

$$\begin{array}{ccc}\hfill \phi \left(\vec{r}\right)& =& \frac{⟨\lambda ⟩r_e}{⟨{\sigma }_{\mathrm{KN}}⟩}\ln \{{[1-(⟨{\lambda }^2/{\sigma }_{\mathrm{KN}}⟩R_2{r}_e{\nabla }^2/2\pi M)]}^{-1}\hfill \\ & & ·(M^{2}I\left({\vec{r}}_D\right)/I_{\mathrm{in}})\}\hfill \end{array}$$ (7)

Here, the operator ∇² denotes the two-dimensional transverse Laplacian differential operator ∇² ≡ (∂²/∂x² + ∂²/∂y²) and the operator $\mathcal{D}\left({\nabla }^2\right)\equiv {\left[1-\left(⟨{\lambda }^2/{\sigma }_{KN}⟩R_2{r}_e{\nabla }^2/2\pi M\right)\right]}^{-1}$ is a pseudodifferential operator. The action on a function $g\left(\vec{r}\right)$ of a pseudodifferential operator such as D (∇²) is defined as

$$\begin{array}{ccc}\hfill \mathcal{D}\left({\nabla }^2\right)g\left(\vec{r}\right)& \equiv & \int \int \exp (i2\pi (\vec{s}-\vec{r})·\vec{f})·\mathcal{D}\left(-4{\pi }^2{\vec{f}}^2\right)\hfill \\ & & ·g\left(\vec{s}\right)d^2\vec{s}·d^2\vec{f},\hfill \end{array}$$ (8)

where $\vec{s}$ and $\vec{f}$ denote the integral variables in the transverse coordinate space and frequency space, respectively. We call Eq. 7 the PAD-based phase retrieval formula.²¹ Recently, we have theoretically and experimentally demonstrated that the PAD-based phase retrievals are robust against the noise, in a stark contrast to the TIE-based method that is unstable in the presence of noise in images.²² Of course, in order for breast tissues to fulfill the phase-attenuation duality condition, the tube voltages should be set to as high as 120–150 kVp, grossly different from the tube voltages employed in current mammography, where tube voltages used are lower than 40 kVp. Although the attenuation-based tissue contrast gets much reduced at 120–150 kVp, the phase-contrast extracted from the phase retrieval will compensate for the attenuation-contrast loss associated with the use of high tube voltages.²² Putting above discussion together, we propose that it is feasible to measure the projected electron densities of a breast by using the high-energy x-ray phase-sensitive imaging technique. One can retrieve the phase map of a breast by using only a single phase-sensitive projection of the breast and performing the PAD-based phase retrieval as is shown in Eq. 7. From the retrieved phase map of the breast one can determine its volumetric breast density by using Eq. 3.

In order to numerically validate the proposed new method of breast density measurement, we constructed a digitally simulated anthropomorphic breast phantom. The phantom includes the simulated breast ductal system, the lobule system, and three masses simulating the invasive ductal carcinoma, and four groups of small calcification spheres (in sizes of 0.54, 0.4, 0.32, and 0.2 mm) simulating the breast microcalcifications. In the breast phantom each voxel was randomly assigned with its electron density ρ_e according to a random Gaussian process with a zero mean and unit standard deviation. To simulate the breast tissue texture, the spatial power spectrum of the random electron density distribution was further modified such that it varies as $|\vec{f}{|}^{1.35}$ for a frequency $\vec{f}$. The resulted phantom's intrinsic by construction volumetric breast density, which we denote by VBD_True, can be computed from the assigned voxel electron densities {ρ_e(v), v = 1, …, N} as

$${\mathrm{VBD}}_{\mathrm{True}}=\frac{\frac{1}{N}\sum _{v=1}^N{\rho }_e\left(v\right)-{\rho }_{e,\mathrm{ad}}}{{\rho }_{e,\mathrm{fg}}-{\rho }_{e,\mathrm{ad}}},$$ (9)

where N is the total number of the phantom voxels.

In order to numerically validate the phase-based method of breast density measurement, we first computed the breast projections with different photon energies. In the simulations we assumed a detector of 50 μm pixels, and set R₁ = R₂ = 1 m. The source's focal spot was assumed as a point source. This is an adequate approximation as long as the blur caused by the focal spot is less than the detector pixel size.¹⁵ In our simulation the magnification factor M = 2 and pixel size is a 50 μm, so a focal spot of 50 μm or smaller in size can be approximately treated as a pointlike source in the simulation. In this simulation we assumed that the source was with a tungsten target and operated at 140 kVp. We assumed that the detection energy spectrum ranges from 54 keV to 140 keV, including the tungsten characteristic radiations. Selecting this photon-energy range in the simulation was to satisfy the phase-attenuation duality condition with good approximation. This implied that the source should operate at high voltages and with heavy filtrations in the implementation of the proposed method. For a given photon energy, a ray-tracing algorithm was used to compute the ray integrals such as ${\int }_{\mathrm{ray}}{\rho }_e(\vec{r},s)ds$ in the projection. The x-ray phase-shifts and attenuations through the phantom were determined as ${\phi }_{\mathrm{ph}}\left(\vec{r}\right)=-\lambda r_e{\int }_{\mathrm{ray}}{\rho }_e(\vec{r},s)ds$ and $A_{\mathrm{ph}}^2\left(\vec{r}\right)=\exp \left[-{\sigma }_{\mathrm{KN}}\left(E\right){\int }_{\mathrm{ray}}{\rho }_e(\vec{r},s)ds\right]$, as we assumed that the phase-attenuation duality condition is satisfied for fibroglandular and adipose tissues in this simulation to numerically validate the proposed method. Using the computed ${\phi }_{\mathrm{ph}}\left(\vec{r}\right)$ and $A_{\mathrm{ph}}^2\left(\vec{r}\right)$, we simulated the x-ray Fresnel diffraction of the exit x rays propagating from the breast phantom to the detector,²³ and thereby we computed the detected Fresnel diffraction intensity $I_{\mathrm{mono}}({\vec{r}}_D,E)$ for each individual energy E. To compute the phase-sensitive projection intensity $I_{\mathrm{poly}}\left({\vec{r}}_D\right)$ with the polychromatic x rays, we summed the $I_{\mathrm{mono}}({\vec{r}}_D,E)$ over photon energies in the detection spectrum. On the other hand, it is important to examine the effects of the noise on the accuracies in the breast density measurement. In fact, the robustness of the phase retrieval method is critical to the success of the proposed phase-based method of breast density measurement, as some methods in the literature may be unstable at some substantial noise levels.²² To simulate the noise effects, we added the Poisson noise with different noise-levels into the simulated phase-sensitive projections of the phantom, following the standard noise simulation method used in the literature.²⁴ Employing the PAD-based phase-retrieval formula 7, we retrieved the phase map $\phi \left(\vec{r}\right)$ of the breast phantom. Using the retrieved $\phi \left(\vec{r}\right)$ and Eq. 3 we reconstructed the phantom's volumetric breast density VBD.

RESULTS

According to the aforementioned method, we constructed a digitally simulated anthropomorphic breast phantom, which simulates a compressed breast of 4.5 cm thick, 20 cm in lateral, and 10 cm in chest wall-nipple dimension. Applying Eq. 9 to the randomly assigned electron densities of this phantom, we found the phantom's intrinsic by construction volumetric breast density VBD_True = 0.5025. We take this VBD_True value as the reference to evaluate the accuracy of the reconstructed phantom volumetric breast density.

From the simulated phase-sensitive projection image of the phantom and using Eqs. 7, 2 with T_c = 4.5 cm, we computed the $m\left(\vec{r}\right)$-image of the breast phantom, the map of the phantom's fibroglandular-tissue fractions, as is shown in Fig. 1. In this image, the shade of gray of a pixel represents the computed fraction of fibroglandular tissues along a ray passing the pixel. A lighter shade of gray represents a larger fraction of fibroglandular tissues. In this way, the $m\left(\vec{r}\right)$-image exhibited good breast tissue contrast as well (Fig. 1.) Moreover, this $m\left(\vec{r}\right)$-image provided the basis for reconstructing the phantom's volumetric breast density. Using the VBD-formula 3, we found the phantom's volumetric breast density VBD = 0.5028, which differs from its intrinsic volumetric breast density by only 0.06%, achieving an excellent accuracy. Furthermore, testing the breast density determination from the projection images with different signal-to-noise-ratio (SNR) levels associated with different mean glandular doses. The SNR, which is similarly defined as in conventional mammography, is defined as the mean intensity over a ROI drawn in the center of the phantom projection image over the standard deviation of the intensity values. We tested the cases with the projection SNR = 34, 64, and 136, which are corresponding to estimated mean glandular doses of 0.14, 0.57, and 2.21 mGy, respectively. We found that the reconstructed VBD values keep achieving good accuracies, in spite of the presence of noise in the simulated projections. With the projection SNR = 34 we found the reconstructed VBD = 0.5115, which differs from the intrinsic volumetric breast density VBD_True by only 1.79%. The error in reconstructed VBD decreases with increasing projection SNR value. For example, we found the reconstructed VBD = 0.5106 for SNR = 68, which differs from VBD_True by only 1.61%. The error in VBD is further reduced to 1.55% for SNR = 134. All these results numerically validated the proposed phase-shifts-based method as a novel breast density measurement method with good accuracies.

Figure 1.

The computed $m\left(\vec{r}\right)$-image of the breast phantom, namely, the map of the phantom's fibroglandular-tissue fractions.

DISCUSSION AND CONCLUSIONS

Current x-ray-based methods of breast density measurement are all based on small attenuation differences between breast tissues. However, as the linear attenuation coefficients of breast tissues vary in a complicated way as the employed x-ray spectrum changes, all these methods have to require elaborated phantom-based system calibrations, some of them include simultaneous exposures of the phantom alongside the breast for system calibrations.^{7, 8, 10} The proposed phase-shifts-based method measures the tissue projected electron densities rather than their projected attenuation coefficients. Our strategy of direct probing tissue electron densities simplifies the system calibration for breast density measurement. In fact, as Eqs. 3, 7 involve the spectral averages of x-ray wavelengths and the Klein-Nishina cross sections, the system parameters that our method requires are the source x-ray spectra and detector responses. Another key idea in our method is to use the PAD-based phase retrieval technique for obtaining the breast phase map. As breast imaging is stringently regulated for radiation doses involved, there are always substantial noises in breast projection images. Some common phase retrieval methods in the literature are actually unstable in presence of noise,²² and the large errors in the retrieved phase map caused by the retrieval instability would spoil the breast density measurement. The high robustness against noise of the PAD-based phase retrieval technique is crucial to our method for achieving good accuracies in breast density determination. Of course, many future works are required for implementing this phase-based method of breast density measurement. In our compressed breast model, 13.5% of breast volume is in the curved periphery zone of the breast, and breast thickness in the periphery is not uniform and falls off quickly. This portion of tissues was not included in the VBD computation for avoiding the errors in thickness estimation. Therefore, if the periphery has a quite different breast density than the rest of breast, this approach may cause errors in the computed VBD. Hence, how to accurately estimate the breast thickness is an area for improvement, and it is also an active research area in implementing other x-ray imaging-based methods of breast density measurement.²⁵ In addition, it will be useful to develop a correction method for the VBD errors in cases where an imaging setup does not accurately satisfy the phase-attenuation condition.

In summary, in this work we propose for the first time an x-ray phase-shifts-based method of measuring the volumetric breast density. The proposed Eq. 3-based method of breast density measurement can be applied to any of phase-sensitive imaging techniques such as the inline or the grating-based techniques. This phase-shifts-based method measures the tissue projected electron densities rather than their projected attenuation coefficients. The direct probing of tissue electron densities in our method simplifies the system calibration requirement. The robust PAD-based phase retrieval method employed in the inline technique ensures the accuracy of the reconstructed volumetric breast densities. Constructing an anthropomorphic digitally simulated breast phantom, we performed simulations for validating this novel method of breast density measurement. Testing with different projection SNR-levels in the simulations, we found that the reconstructed volumetric breast densities achieved good accuracies compared to the phantom's intrinsic volumetric breast density. All these results numerically validated the proposed phase-shifts-based method as a novel method of breast density measurement with good accuracies.