Stochastic Numerical Breast Phantom Generator

• Stochastic numerical breast phantom generation is a computational framework that produces ensembles of digital breast models exhibiting anatomical and tissue property variability.
• It enables repeatable virtual imaging trials, reconstruction benchmarking, observer studies, and dosimetry by simulating controlled variations in anatomy, pathology, and modality-specific physics.
• The methodology integrates surface-based modeling, voxelized procedural generation, image-derived phantom construction, and deep learning techniques to achieve realistic and task-valid simulations.

A stochastic numerical breast phantom (NBP) generator is a computational framework that produces ensembles of digital breast models with controlled variability in anatomy, tissue composition, pathology, and modality-specific physical properties, so that virtual imaging trials, reconstruction studies, observer studies, education, and dosimetry can be performed under repeatable and clinically relevant conditions. In the literature, NBPs appear as surface-based compression models for virtual mammography, voxelized anthropomorphic phantoms adapted from the FDA VICTRE project, MRI-derived optical phantoms, and measurement-aware generative models that translate mathematical phantoms into realistic mammographic appearances. A consistent requirement across these formulations is that the ensemble display the variability in anatomy and object properties that is representative of the to-be-imaged patient cohort (Li et al., 2021, Xu et al., 2024).

1. Historical emergence and conceptual scope

An early explicit step toward numerical breast phantom generation for mammography was the development of a computational model that reproduces the breast compression processes used to obtain the mammogram by means of the Surface Evolver. That work treated the breast as a surface enclosing a region, allowed the addition of structures representing different tissues, muscles and glands, and combined simulation with laboratory tests using phantoms. The reported outcome was that the model reproduced the same shapes and measurements really taken from the volunteer’s breasts, while its modular input—values taken by a simple tape-measure together with the mammographies—was identified as a basis for potentially stochastic or automatic generation of a wide variety of breast phantoms (Nascimento et al., 2014).

Subsequent work broadened the scope of NBP generators beyond virtual mammography. In ultrasound computed tomography (USCT), three-dimensional stochastic numerical breast phantoms were introduced to enable clinically relevant virtual imaging trials, with explicit modeling of breast size, composition, acoustic properties, tumor locations, and tissue textures (Li et al., 2021). In quantitative optoacoustic tomography (qOAT), a virtual imaging framework extended a stochastic numerical breast phantom generator by incorporating skin tone variation and both benign and malignant lesions, and by linking the phantom generator to end-to-end simulation of optical and acoustic measurement formation (Park et al., 30 Sep 2025). Parallel developments in fluorescence molecular tomography (FMT), microwave imaging, and breast CT dosimetry demonstrate that “NBP generator” is best understood not as a single algorithmic family but as a class of modality-coupled stochastic object models whose realism is defined relative to a downstream imaging task (Zhu et al., 2018, Ambrosanio et al., 2021, Entezam et al., 6 May 2026).

This breadth has methodological consequences. Some generators are primarily anatomical and geometric; some are primarily physical-property generators; some are image-domain generators intended for human interpretation or algorithm training; and some are explicitly measurement-aware stochastic object models. The literature therefore treats the generator not merely as a source of synthetic anatomy, but as a formal interface between object variability and imaging-system design.

2. Core modeling paradigms

Surface-based modeling remains one foundational paradigm. In the Surface Evolver formulation, the breast is represented as a triangulated surface layer to which physical properties such as mass, gravity, and pressure are assigned, while bulk elasticity is not directly simulated. Instead, equilibrium configurations are approximated by energy minimization, specifically through the Willmore functional,

$\int_M H^2 \, dS,$

where $M$ is the surface, $H$ the mean curvature, and $dS$ the surface area element. This supports transitions among anatomically and clinically relevant positions, including SRG, STU, LAT, CRC, LET, and MLO, with calibration by a “virtual tape-measure” within the simulation (Nascimento et al., 2014).

A second paradigm is voxelized anthropomorphic generation based on procedural anatomy. For USCT, an existing stochastic 3D breast phantom from the FDA VICTRE project was extended and adapted by stochastic, multi-parametric deformations of base superquads, with shape parameters such as $a_{1t}, a_{1b}, a_{2r}, a_{2l}, a_3$ and deformation parameters $\epsilon_1, B_0, B_1, H_0, H_1$ sampled from Gaussian or truncated Gaussian distributions set from empirical breast CT data. Tissue labels were then adapted to retain the tissues visible to USCT, lesions were inserted, and acoustic properties were assigned stochastically per tissue class (Li et al., 2021). OpenBreastUS used a related VICTRE-based procedural tool to generate 8,000 anatomically realistic human breast phantoms, again relying on truncated Gaussian sampling of shape-defining parameters and stochastic control of the adipose fraction through targetFatFrac for the four ACR BI-RADS density categories (Zeng et al., 20 Jul 2025).

A third paradigm is image-derived phantom construction. In FMT, a digital breast phantom was generated from an MR database and segmented into blood vessels, skin layer, fat, and fibroglandular tissues, after which tissue-specific optical properties were assigned and tumors were inserted digitally (Zhu et al., 2018). In propagation-based phase-contrast breast CT dosimetry, voxel-based anthropomorphic breast phantoms were derived directly from synchrotron BCT images of freshly excised human mastectomy specimens; segmentation and HU-based tissue classification yielded heterogeneous patient-specific phantoms, material-homogeneous variants, homogeneous phantoms, and stochastic homogeneous phantoms generated synthetically for parameter studies (Entezam et al., 6 May 2026).

A fourth paradigm uses generative models. MammoGANesis employed StyleGAN2 to synthesize $512 \times 512$ mammograms with stochastic latent variation and controllable editing in latent and activation spaces (Zakka et al., 2020). AmbientCycleGAN addressed a different problem: establishing realistic and interpretable stochastic object models by translating parameterized mathematical phantoms, specifically clustered lumpy background models, into realistic synthetic breast images using noisy mammographic measurements and a known measurement operator $\mathcal{H}_n$ (Xu et al., 2024). This suggests that modern NBP generation includes both explicit anatomical simulation and learned domain translation.

3. Sources of stochasticity and controllable variability

Stochasticity in NBP generators is introduced at multiple levels. In procedural anatomical generators, randomness appears first in global breast geometry. For example, OpenBreastUS sampled $$a1b, a1t, a2l, a2r \sim \mathcal{TN}(5.0, 2.0, 3.5, 7.5)$$ and $a3/a1b \sim \mathcal{TN}(1.4, 0.1, 1.0, 1.5)$, while additional parameters such as SkinScale, SkinScaleNippleDir, and skinStrength were randomized to simulate different skin thickness and elasticities (Zeng et al., 20 Jul 2025). In the microwave phantom generator, the outer breast contour began from an ellipse with axes uniformly sampled in $M$0 cm, randomized center placement within a 1 cm radius circle, random orientation in $M$1, and skin layer thickness in $M$2 mm (Ambrosanio et al., 2021).

Randomness then enters internal tissue composition. VICTRE-based USCT phantoms relabeled tissue maps so that fat, glandular, skin, and ligament remained, after which speed-of-sound, density, and acoustic attenuation were sampled from predefined distributions per tissue type. Sub-tissue heterogeneity was added through Gaussian random fields in fatty and glandular regions, and attenuation was modeled by the power law

$M$3

These steps create ensembles of piecewise-constant acoustic property maps with stochastic anatomical region boundaries, values, and within-tissue texture (Li et al., 2021). In microwave imaging, the interior was generated by a stochastic 2D multifractal random field, thresholded into fibro-glandular, transitional, and adipose classes, and then assigned dielectric properties sampled from empirical distributions with spatial correlation between neighboring pixels (Ambrosanio et al., 2021).

Pathological and physiological variability are likewise encoded stochastically. In the qOAT framework, tissues included fat, skin, glandular, nipple, muscle, ligament, ducts, arteries, and veins, while functional properties such as total hemoglobin, blood oxygen saturation $M$4, and chromophore volume fractions $M$5, $M$6, $M$7, and $M$8 were sampled within physiological ranges derived from the literature. Optical absorption was computed as

$M$9

and initial pressure by

$H$0

The same framework modeled no lesions, malignant tumors with viable tumor cell region, necrotic core, and angiogenesis region, and benign lesions including fibroadenomas and simple cysts (Park et al., 30 Sep 2025).

Skin tone is an additional stochastic dimension in optoacoustic phantom generation. A two-layer skin model incorporated melanosome volume fraction $H$1 in the epidermis-included layer, with $H$2 sampled according to Fitzpatrick phototype; for phototype VI, the reported range was $H$3. This directly changed optical absorption and therefore superficial light deposition and deep fluence (Park et al., 30 Sep 2025). In phase-contrast CT dosimetry, incomplete native skin coverage in excised specimens was handled by generating synthetic skin layers of 1, 2, or 3 mm thickness through morphological dilation, voxel values sampled from a normal distribution matched to real skin statistics, and Gaussian smoothing (Entezam et al., 6 May 2026).

A recurring theme is controllability. Mathematical phantoms and parameterized procedural models offer explicit control over geometry, tissue fractions, lesion presence, and physical properties. Generative approaches seek to recover that control without sacrificing realism. AmbientCycleGAN preserved interpretable manipulation because modifying the number, position, or density of lumps or clusters in the input mathematical phantom yielded corresponding localized changes in the synthesized output (Xu et al., 2024). MammoGANesis offered both global editing, by principal component traversal in $H$4-space, and local editing, through spherical $H$5-means clustering of early generator activations (Zakka et al., 2020).

4. Modality-specific realizations

Modality	Generator basis	Representative use
Mammography	Surface Evolver surface model	External compression simulation
DBT	Bakic and XCAT breast phantoms	System optimization in VITs
USCT	VICTRE-derived stochastic 3D acoustic phantoms	Reconstruction development and benchmarking
qOAT	SOA-NBP with lesion and skin-tone modeling	End-to-end system design evaluation
FMT	MRI-derived segmented digital phantom	Reconstruction-method comparison
Microwave imaging	Ellipse plus multifractal tissue generator	Neural-network training
Synchrotron BCT	Patient-specific voxel phantoms from BCT	Monte Carlo dosimetry

In digital breast tomosynthesis (DBT), phantom choice itself becomes an experimental variable. One study compared the Bakic phantom, described as a stochastically generated, fully parameterizable, mathematical/anthropomorphic phantom with 0.2 mm isotropic voxel size, and the XCAT phantom, described as a numerically compressed, anthropomorphic phantom created from patient breast CT data with 0.25 mm isotropic voxel size (Kavuri et al., 2024). In this setting, the NBP generator is not merely a source of anatomy but a determinant of optimization outcomes.

In USCT, stochastic phantoms are tightly integrated with acoustic forward models. The VICTRE-derived acoustic phantoms were used with a k-space, time-domain pseudo-spectral wave solver and a simulated USCT system comprising 1024 transducers in a ring, while the public release included 52 sets of 2D slices and simulated RF data and 4 full 3D phantoms, each at 0.1 mm voxel size (Li et al., 2021). OpenBreastUS scaled this paradigm to 8,000 anatomically realistic phantoms and over 16 million frequency-domain wave simulations using real USCT configurations, thereby turning the phantom generator into an infrastructure for benchmarking neural operators in forward and inverse wave imaging (Zeng et al., 20 Jul 2025).

In optical and optoacoustic imaging, the generator must support chromophore or fluorophore modeling. The MRI-derived FMT phantom used 40 laser sources and 40 detectors arranged around the breast model, with two tumors of size 1.6 cm inserted at varying center-to-center separations and fluorophore concentration assumed exclusively in the tumor regions (Zhu et al., 2018). The qOAT framework extended this principle to multi-wavelength fluence maps, initial pressure maps, and realistic OAT voltage measurements including transducer spatial impulse response and acousto-electric impulse response (Park et al., 30 Sep 2025).

In educational mammography synthesis, MammoGANesis was trained on 162,988 real mammograms, with 152,973 for training and 10,015 for validation, across all standard mammographic views and ages 18+ (Zakka et al., 2020). Although these are generated images rather than full physical-property phantoms, the reported stochastic variation in adipose tissue, calcification, and lesion-related features places the method in the broader NBP landscape when the target task is image interpretation rather than physics-based simulation.

5. Validation, realism, and task dependence

Validation strategies differ substantially across the literature, but a consistent pattern is that realism must be established relative to the intended task. The Surface Evolver mammography model was validated against real volunteer measurements and laboratory compression tests with transparent physical breast phantoms. During the main postural transitions from SRG to STU to LAT, the simulation reproduced a decrease in breast volume from roughly 700 cm³ to 680 cm³ to 660 cm³, mirroring real anatomical behavior (Nascimento et al., 2014).

For DBT, realism was assessed not only visually but through structural frequency content and human observer performance. Noise power spectrum (NPS) analysis showed NPS slope $H$6 for Bakic phantoms and $H$7 for XCAT phantoms. Human observer localization ROC studies for 3-mm lesion detection found peak performance at approximately 2.5 degrees between projections in Bakic phantoms and approximately 6 degrees in XCAT phantoms, with inter-observer intraclass correlation 0.92 to 0.95. Several gaze metrics correlated strongly with AUC, including first hit time with $H$8 and lesion dwell time or number of fixations on lesion with approximately $H$9. The reported conclusion was that system optimization outcomes from virtual imaging trials can vary with phantom types and structural frequency components (Kavuri et al., 2024).

Generative models require different evidence. MammoGANesis evaluated realism by a double-blind study with four expert mammography radiologists, yielding an average AUC of 0.54 in classifying real versus generated images and an average of 5.75 rounds before 3 errors in a six-image discrimination task (Zakka et al., 2020). AmbientCycleGAN was evaluated by Fréchet Inception Distance, radially averaged power spectrum, SSIM distributions, and a Hotelling observer detection task. Reported FID values were 13.91 for Opex-CLB to Simpiso-CLB and 17.89 for Opex-CLB to real mammograms, compared with 127.69 and 82.23 for CycleGAN, while ROC performance on synthesized backgrounds nearly matched the ground truth in the signal-detection study (Xu et al., 2024). These results distinguish visual plausibility, structural fidelity, and task-based validity as separable criteria.

Dosimetric validation further sharpens this point. In synchrotron phase-contrast breast CT, mean glandular dose depended strongly on anatomy and energy; higher glandular density reduced MGD, larger breast volume increased dose, and a 2 mm increase in skin thickness reduced MGD by 10%. Homogeneous phantoms underestimated DgN compared to patient-specific phantoms by up to 36% for thin skin, whereas material-homogeneous phantoms reduced the DgN error to 3–5%, indicating that spatial distribution dominates dosimetric impact (Entezam et al., 6 May 2026). A plausible implication is that anatomical realism is not a cosmetic property of an NBP generator but a determinant of quantitative inference.

6. Open resources, limitations, and current directions

NBP generators increasingly appear as open research infrastructure. The Surface Evolver mammography tool was implemented as a user-accessible program available via a virtual machine and based on open-source software including Surface Evolver and Geomview (Nascimento et al., 2014). The USCT phantom framework released examples of breast phantoms and 52 sets of simulated measurement data, together with a Python package implementing generation, stochastic parameter sampling, acoustic property map creation, texture synthesis, lesion modeling, and file export (Li et al., 2021). OpenBreastUS made its dataset and associated code publicly available and positioned the collection as a benchmarking platform for neural PDE solvers (Zeng et al., 20 Jul 2025). The qOAT framework made 1,020 NBPs and associated measurement data publicly available to support research in optoacoustic and optical imaging (Park et al., 30 Sep 2025).

At the same time, the literature identifies clear limitations. Surface Evolver works only with surface representations and idealizes internal material as liquid or gas rather than solid tissue; precise mechanical properties of real tissue are not fully modeled, and validation phantoms are not perfectly biomimetic in elasticity (Nascimento et al., 2014). In MammoGANesis, the current $dS$0 resolution may miss some details present in clinical images, such as small microcalcifications or subtle tissue structures, and continuous latent spaces can produce ambiguous “in-between” states for discrete attributes (Zakka et al., 2020). In DBT virtual imaging trials, lack of standardization in phantom structure and frequency content undermines reproducibility and comparability of optimization studies, motivating the recommendation that researchers analyze and report NPS and structural realism for phantom ensembles (Kavuri et al., 2024).

A broader methodological tension concerns realism versus interpretability. Purely mathematical stochastic object models such as clustered lumpy backgrounds provide explicit control but may not comprehensively capture realistic object variations, whereas models trained directly on imaging data may better match the appearance of true tissues but often lack interpretable and parameterized control. AmbientCycleGAN was proposed specifically to bridge that gap by translating mathematical stochastic object models to realistic stochastic object models using noisy measurement data (Xu et al., 2024). This suggests that a central current direction in NBP generation is not simply making phantoms more realistic, but making them simultaneously realistic, task-valid, and controllable.

Across modalities, the stochastic NBP generator has therefore evolved from a phantom-production utility into a formal experimental instrument. It encodes assumptions about anatomy, tissue physics, pathology, measurement formation, and observer or algorithmic tasks. Its design directly affects reconstruction benchmarking, acquisition optimization, educational realism, and patient-specific dosimetry. In that sense, the generator is now part of the imaging system being studied, rather than merely an input to it.