# A Novel 3D Stochastic Solid Breast Texture Model for X-Ray Breast Imaging

## Abstract

Performance assessment of breast x-ray imaging systems through clinical imaging studies is expensive and may result in unreasonable high radiation doses to the patient. As an alternative, several research groups are investigating the potential of virtual clinical trials using realistic 3D breast texture models and simulated images from those models. This paper describes a mathematically defined solid 3D breast texture model based on the analysis of segmented clinical breast computed tomography images. The model employs stochastic geometry to mimic small and medium scale fibro-glandular and adipose tissue morphologies. Medium-scale morphology of each adipose compartment is simulated by a union of overlapping ellipsoids. The boundary of each ellipsoid consists of small Voronoi cells with average volume of 0.5 mm^{3}, introducing a small-scale texture aspect. Model parameters were first empirically determined for almost entirely adipose breasts, scattered fibro-glandular dense breasts and heterogeneously dense breasts. Preliminary evaluation has shown that simulated mammograms and digital breast tomosynthesis images have a reasonable realistic visual appearance, depending though on simulated breast density. Statistical inference of model parameters from clinical breast computed tomography images for the variety of fibro-glandular and adipose tissue distributions observed in clinical images is ongoing.

### Keywords

Solid 3D breast texture model Stochastic geometry## 1 Introduction

Since clinical imaging studies are expensive and time consuming, there has been an increased research investment into less costly and more time-efficient simulation studies [1, Sect. 12.5]. Reliable simulation studies in the field of digital breast tomosynthesis (DBT), and more generally in 3D imaging of the breast, require a realistic numerical 3D anthropomorphic test object model, a realistic numerical image acquisition chain and task-based mathematical observer models.

Several numerical 3D anthropomorphic breast object models have been proposed [1, 2, 3, 4, 5]. Each model presents advantages and restrictions in terms of the visual realism of breast object images, the consistency between statistical features of test object and clinical images, and the potential to mimic the anatomical variability seen in real breast images. These aspects may impact the model validity for certain performance assessment studies.

Pokrajac *et al.* [1, Sect. 12.3.1] and Mahr *et al.* [4] have proposed 3D breast phantoms with mathematically defined structures. To simulate medium scale adipose tissue compartments in the fibroglandular tissue, Pokrajac *et al.* employs a region growing scheme implemented as a Voronoi diagram with a specific norm associated with ellipsoidal shapes [1, Sect. 12.3.1]. Each adipose compartment is simulated as a single geometry controlled by a seed point. Simulation of small scale fibroglandular tissue is obtained by sub-compartmentalization of the adipose tissue compartments [2]. To simulate the medium scale adipose tissue in the fibroglandular tissue, Mahr et al. [4] uses uniformly distributed ellipsoids pointing towards the nipple with stochastic tilt-angles and the semi-axes lengths of the ellipsoids are uniformly distributed and empirically determined. Clusters of 0.5 to 1 mm diameter randomly distributed spheres are positioned at the end-point of each ductal tree terminal branch to mimic small scale structures in the fibro glandular tissue.

We propose an alternative 3D solid breast texture model inspired by the morphology of medium and small scale fibro-glandular and adipose tissue observed in clinical breast computed tomography (bCT) images (UC Davis database). The model admits a mathematical formulation using stochastic geometry theory, a branch of mathematics studying the distribution of random geometric objects. Each adipose compartment is modeled as a union of overlapping ellipsoids and the whole model is formulated as a spatial marked point process [6]. A small-scale texture aspect is introduced by replacing the smooth ellipsoid boundaries by Voronoi cells with average volume of 0.5 mm^{3}. The texture model allows for simulation of visually realistic 2D and 3D breast x-ray images and for providing a complete mathematical formulation.

## 2 Methodology

### 2.1 Materials and Input

*et al.*[2] already illustrated irregular variations around adipose compartments in microscopic breast histological images.

### 2.2 Model Construction Algorithm

- 1.
Given a voxelized texture cube \( W \) with voxel size \( \nu \), uniformly generate \( N \) points, with \( N \) following a Poisson distribution of parameter \( \lambda_{0} \left| W \right| \), where \( \left| . \right| \) denotes the volume measure and \( \lambda_{0} \) is a positive real number. This generates a realization of a homogeneous Poisson point process [6].

- 2.
Generate a Voronoi diagram [6] using points obtained from step 1 as cell centers.

- 3.
Independently from step 1 and step 2, create another set of points by drawing a realization of a spatial point process \( \Phi _{\text{s}} \) in \( W \) with distribution \( {\mathbf{P}}_{s} \). This set of points is named the seed points.

- 4.
For each seed point \( i \), assign a random ellipsoid \( {\text{Ell}}_{i} (\theta ) \) with parameter vector \( \theta = (\varphi_{1} ,\varphi_{2} ,\varphi_{3} , La,Lb,Lc) \) having distribution \( \Theta \), where \( \varphi_{1} ,\varphi_{2} ,\varphi_{3} \) are angles determining the orientation of the ellipsoid and \( La,Lb,Lc \) are the lengths of semi-axes. The ensemble of the seed point process and the ellipsoid parameters \( \Xi = \left\{ {\Phi _{\text{s}} ,\theta } \right\} \) can be treated as a spatial marked point process [6] determined by the joint distribution of \( {\mathbf{P}}_{s} \) and \( \Theta \).

- 5.
Check each point \( x \) of step 1, if \( x \in \bigcup\nolimits_{i} {{\text{Ell}}_{i} } \), in other words, if it falls inside at least one ellipsoid, then assign all voxels of its corresponding Voronoi cell a value 0. Assign all other voxels value 1.

To demonstrate the impact of the irregular adipose compartment boundaries using small Voronoi cells on improved visual realism of simulated images, texture simulations were performed with smooth and irregular ellipsoid boundaries. Smooth boundaries were simulated by starting directly from step 3 and by modifying step 5 through simply assigning each time a voxel the value 0 if it falls inside at least one ellipsoid.

## 3 Simulation Results

*i.e.*all points are independently uniformly distributed). Parameters were empirically determined for almost entirely adipose breasts, scattered fibro-glandular dense breasts and heterogeneously dense breasts (Table 1). Figure 3 shows examples of regions of interest of simulated texture volume slices, mammograms and DBT reconstructed slices from realizations of 3D solid texture model for breasts of different densities. This figure also shows images created using segmented bCT datasets as reference. Mammograms and DBT projection images were simulated by virtually projecting the texture model using the previously described breast x-ray imaging simulator. DBT reconstructed slices were constructed by processing the projections using an in-house iterative reconstruction algorithm. Breast texture deformation to mimic breast compression during mammography and DBT image acquisition was not modeled. Preliminary evaluations on regions of interest of simulated mammograms and DBT reconstructed slices, similarly to those shown in Fig. 3, indicate a high visual realism compared to real clinical images.

Model parameters used for the simulations shown in Fig. 3.

Parameters | Almost entirely adipose | Scattered fibro-glandular dense | Heterogeneously dense |
---|---|---|---|

\( W \) | (5 cm) | ||

\( \lambda_{0} \) | 2 mm | ||

\( \Phi _{\text{s}} \) intensity | 3.5 × 10 | 5 × 10 | 2.3 × 10 |

\( La,Lb,Lc \) (in mm) | \( \mu_{La} \sim N\left( {7.4,1.5} \right) \)\( \mu_{Lb} ,\mu_{Lc} \sim N\left( {4.5,0.7} \right) \) | \( \mu_{La} \sim N\left( {6.4,1.5} \right) \)\( \mu_{Lb} ,\mu_{Lc} \sim N\left( {2.5,0.7} \right) \) | \( \mu_{La} \sim N\left( {6.4,1.5} \right) \)\( \mu_{Lb} ,\mu_{Lc} \sim N\left( {2.5,0.7} \right) \) |

\( \varphi_{1} ,\varphi_{2} ,\varphi_{3} \) | Gaussians with means determined by a vector from the ellipsoid center to a pre-defined nipple. Standard deviations are \( \frac{\pi }{12} \) |

## 4 Realism Assessment Through Psycho-Physical Studies

A two-alternative forced choice (2AFC) experiment was performed to assess the visual realism of simulated DBT reconstructed images. Pairs of 3.5 × 3.5 cm^{2} ROIs, extracted from images simulated from our texture model and clinical bCT data were displayed side-by-side. Images on the left always came from the clinical bCT data set, while images on the right had 50 % chance to be from the texture model and 50 % chance to be from clinical bCT data. The reader had to tell whether the image on the right was from the clinical bCT data set or from the texture model. Similar level of glandular density was maintained for each image pair. In total, 144 image pairs were presented; 52 pairs from almost entirely adipose breasts, 56 pairs from scattered fibro-glandular dense breasts and 36 pairs from heterogeneously dense breasts. Each image pair was displayed during 5 s, followed by the display of a uniform gray–level image for another 5 s; the readers were thus imposed to make a decision within 10 s.

A short training session with 10 image pairs of known ground truth was performed before the real experiment with no time constraint. Images were displayed on 5 M pixels grayscale portrait monitors (SMD 21500 G, Siemens AG; Munchen, Germany) at 100 % resolution. The reading distance was set to be one meter.

Four readers participated in the experiment, all GE Healthcare engineers. Reader 1, 2 and 3 have no prior knowledge of our texture model while reader 4 knows the algorithm of the texture model construction. The percentage of correct answers, Pc, was calculated as an indication for the realism of the simulated images from the texture model. Under the null hypothesis that images simulated from the model and clinical bCT images data cannot be distinguished, Pc = 0.5.

Detection rate and p-values for each reader by type of breast (p < 0.05 means the reader was able to differentiate simulated data from real data).

Almost entirely adipose | Scattered fibroglandular dense | Heterogenously dense | All | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

n = 52 | n = 56 | n = 36 | n = 144 | |||||||||

Reader 1 | 31 | 0.60 | p > 0.05 | 39 | 0.69 | p < 0.05 | 24 | 0.66 | p > 0.05 | 94 | 0.65 | p < 0.05 |

Reader 2 | 30 | 0.58 | p > 0.05 | 36 | 0.64 | p < 0.05 | 30 | 0.83 | p < 0.05 | 95 | 0.66 | p < 0.05 |

Reader 3 | 32 | 0.62 | p > 0.05 | 44 | 0.79 | p < 0.05 | 30 | 0.83 | p < 0.05 | 105 | 0.73 | p < 0.05 |

Reader 4 | 39 | 0.75 | p < 0.05 | 51 | 0.86 | p < 0.05 | 32 | 0.89 | p < 0.05 | 118 | 0.82 | p < 0.05 |

## 5 Discussion and Conclusion

In this paper, we described a mathematically defined solid 3D breast texture model based on the analysis of segmented clinical breast computed tomography images. Our model is formulated as a spatial marked point process, exhibiting mathematical formulation.

Preliminary evaluations of simulated 2D and 3D breast x-ray images using a prototype of the solid breast texture model, with empirically determined parameters appear to have high visual realism compared with clinical images. To assess visual realism of synthetic DBT images simulated using our texture model, we performed a 2-AFC experiment involving four readers. This preliminary experiment results demonstrated that the model with empirical parameters can simulate fairly realistic DBT reconstructed images for almost entirely adipose breast types. The results also demonstrate that a quantification artifact, introduced by the size of the Voronoi cells at the transition of adipose/glandular tissue, reduces the realism of the texture model for higher density breasts. Using the empirically determined parameters, our current model is thus restricted to simulate a limited variability of breast tissue types encountered in real breasts. We will further address this limitation by fitting our model parameters to different breast types seen in the clinical bCT database.

Due to the stationarity of the seed point process which assumes no interactions between the ellipsoids, simulations of larger texture volumes results in reduced visual realism in simulated images. To address this, a point process with clustering interactions might be worth investigating.

## Notes

### Acknowledgements

We thank Dr. John Boone, University of Davis, for allowing us to use his large database of clinical bCT images. This study was partially funded by ANRT, under CIFRE convention N° 2013/1052.

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