Image-based reconstruction of anthropomorphic breast phantoms for synthetic mammogram generation

Image-based reconstruction of anthropomorphic breast phantoms for synthetic mammogram generation Martina Oria a,b , Riccardo Ferrero a , Chiara Andreis c , Marta Vicentini a , Ruben van Engen d , Carlijn Roozemond d , Paola Lamberti c , Sara Remogna c , Alessandra Manzin a,* a Istituto Nazionale di Ricerca Metrologica (INRiM), Torino, Italy b Politecnico di Torino, Torino, Italy c Dipartimento di Matematica "Giuseppe Peano", Universit` a degli Studi di Torino, Torino, Italy d Dutch Expert Centre for Screening (LRCB), Nijmegen, Netherlands ARTICLE INFO Keywords: Synthetic mammograms Breast digital phantoms X-ray diagnostic imaging Mammography In silico modelling Image acquisition modelling ABSTRACT The aim of this work is the generation of realistic synthetic mammograms, using as an input of the imaging acquisition simulation process digital anthropomorphic phantoms, reconstructed from sets of dedicated breast computed tomography (BCT) images from different patients. The voxel-based structure and the segmentation into fibroglandular, adipose and skin tissues are performed through trivariate tensor-product B-spline approximation and morphological operations. The obtained phantoms can be modified by means of geometrical transformations that replicate typical breast shape deformities, and by locally introducing virtual masses and calcifications. After simulating biomechanical compression of the 3D breast phantoms, we generate the mammograms in both craniocaudal (CC) and mediolateral oblique (MLO) views, modelling the x-ray interaction with breast tissues with a Monte Carlo approach implemented in the in silico breast imaging pipeline VICTRE. The methodology proposed here can contribute to the creation of synthetic mammogram databases, to be used for in silico testing of diagnostic and therapeutic techniques, as well as for the validation of artificial intelligence (AI) systems in diagnostic imaging and cancer screening. The great advantage is that, from a single BCT scan, it is possible to generate multiple realistic mammograms, with different anatomical features, in terms of breast shape and size, and type and location of lesions. 1. Introduction Anthropomorphic digital phantoms can be a valuable tool for in silico testing and optimization of screening, diagnostic and therapeutic techniques. These enable us to investigate dosimetry implications, as well as how the distribution of tissues and their microstructure can affect the outcomes of different imaging modalities and disease treatments [1–5]. Examples of computational phantoms with realistic anatomical details and high resolution have been developed to support the improvement of diagnostics methods based on magnetic resonance imaging (MRI) [6], ultrasound and photoacoustic imaging [7], x-ray based techniques [8–10], and multi-modal imaging [11]. Moreover, the availability of these tools has enabled the implementation of virtual clinical trials for therapeutic applications, including radiotherapy [12], hyperthermia [13], thermal ablation [14], and surgery [15,16]. To simulate pre-clinical tests and enlarge the spectrum of in silico assessment, animal computational models have also been developed besides anthropomorphic phantoms [17], finding applications in both dosimetry and therapeutic technique testing [18–20]. Another important application of 3D digital anthropomorphic phantoms is the generation of synthetic images by means of physical computational modelling, with the scope of enriching existing medical image datasets, which are usually strongly heterogeneous and not sufficiently populated for specific subgroups. This can allow the creation of databases, which can be used for the robust training and accurate validation of artificial intelligence (AI) systems, giving instruments also for explainability tasks. Great efforts have been made in the field of breast imaging, where an invaluable example is the 3D phantom generator integrated in the in silico breast imaging simulator VICTRE (Virtual Imaging Clinical Trials for Regulatory Evaluation), designed by Food * Corresponding author. E-mail address: [email protected] (A. Manzin). Contents lists available at ScienceDirect Computers in Biology and Medicine journal homepage: www.elsevier.com/locate/compbiomed https://doi.org/10.1016/j.compbiomed.2025.111121 Received 14 April 2025; Received in revised form 17 September 2025; Accepted 19 September 2025 Computers in Biology and Medicine 198 (2025) 111121 Available online 1 October 2025 0010-4825/© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). and Drug Administration (FDA), to replicate the entire acquisition pipeline of mammography and digital breast tomosynthesis [21]. The VICTRE phantom generator enables to create breasts with different volumes and shapes, including anatomical structures like nipple, ductal tree structure with terminal duct lobular units, glandular compartments, fat and glandular lobules, Cooper’s ligament network, and vasculature. To improve realism in tissue texture, advances have been made by deriving the digital breast phantoms from high-resolution 3D clinical breast images, like the ones acquired with the dedicated BCT scanner developed at UC Davis (California, USA). As a fundamental result, a cohort of 150 computerized breast phantoms was generated, to be used for virtual clinical trials in x-ray breast imaging and dosimetry [8]. Different strategies have also been implemented to create a large set of realistic breast phantoms from a relatively small set of BCT data. These include the use of statistical methods exploiting principal component analysis (PCA) [22], or the application of mesh-based geometric transformations and morphing techniques [23]. In this study, we focus on the reconstruction of breast digital phantoms, to be used for the generation of synthetic mammograms with an in silico approach that leverages the VICTRE pipeline [21]. The 3D phantoms are derived from clinical BCT images, performing tissue segmentation by means of trivariate tensor-product B-spline approximation [24], in combination with morphological operations. The semi-automatic segmentation and voxel-structure creation are carried out by distinguishing fibroglandular, adipose and skin tissues. To simulate the entire image acquisition process, the phantoms are virtually compressed to a specified thickness, and then used as an input of the Monte Carlo x-ray simulator implemented in VICTRE. To further enrich the initial dataset and reflect the natural variety of breast morphology and tissue distribution, we modify the original phantoms by (i) applying realistic deformations and (ii) inserting artificial lesions, like masses with well-defined or spiculated margins, and clustered and tubular type calcifications. Synthetic 2D mammograms are then generated in both craniocaudal (CC) and mediolateral oblique (MLO) views. A sample of images is reported, as a proof of the flexibility of the proposed methodology in obtaining, from a low number of reference phantoms, a range of mammograms with large variability in terms of breast shape and size, and type and location of lesions. As a result, from a few sets of BCT images acquired on different patients we can generate 2D mammograms with reduced correlation among each other, and thus useful to expand available datasets for AI system testing and evaluation. 2. Methods In the following, we describe the entire pipeline for the generation of synthetic mammograms, from 3D breast phantom reconstruction and their geometric transformation up to the x-ray absorption simulation. The implemented methodology, whose steps are depicted in the schematic of Fig. 1, enables us to create a 3D reference phantom from a set of images from one patient undergoing dedicated BCT, after the image conversion into a trivariate B-spline solid. This means that N reference phantoms can be obtained from N sets of BCT images acquired on N patients. The N reference phantoms can be transformed by: (i) applying different types of realistic deformations; (ii) inserting lesions (masses or calcifications) at voxel level. Then, through this procedure we can produce several phantoms differing in shape, volume, tissue composition, and lesion type, position, size and morphology. 2.1. Reference phantom generation As an input for the generation of 3D reference phantoms, we use dedicated BCT images obtained from a single scan and with a spatial resolution of 273 μ m along the three directions. The images were acquired at the Radboud University Medical Center, with a dedicated BCT system from Koning Corporation (West Henrietta, NY, USA), setting the x-ray tube with tungsten target and aluminum filter to a voltage of 49 kV; the first half value layer is 1.39 mm Al, and the nominal focal spot is 0.3 mm [25]. Tissue segmentation is performed on the entire breast volume, which is obtained by combining the BCT slices on the coronal view (yz-plane) from muscle to nipple direction (x-axis) (Fig. 2a). The voxelization and the identification of the main tissues (fibroglandular, adipose and skin tissues) are carried out after applying an imaging operator based on trivariate tensor product B-splines [24], which enable us to visualize the complex texture of breast, and to determine interactively the grey-level Fig. 1. Schematic of the implemented methodology, including the conversion of one set of dedicated BCT images acquired on one patient into a trivariate B-spline solid, the assignment of tissue properties after greyscale histogram analysis, the phantom transformation via deformation and lesion inclusion at voxel level, its biomechanical compression, and the generation of the synthetic mammogram, from the simulation of the interaction between x-ray and breast tissues. M. Oria et al. Computers in Biology and Medicine 198 (2025) 111121 2 thresholds for semi-automatic tissue segmentation. In detail, the discretization in voxels is carried out by introducing a 3D grid with resolution higher than that of the BCT images, described by three knot vectors U={u0,u1, ..., unu+pu+1},V={v0,v1, ..., vnv+pv+1}and W={w0, w1, ..., wnw+pw+1}along the u, v and w directions. Then, we consider a trivariate B-spline solid of degree (p u , p v , p w ), which is a tensor product volume defined as V(u,v,w) = ∑ nu i=0∑ nv j=0∑ nw k=0 Ni,pu(u)Nj,pv(v)Nk,pw(w)Pijk.(1) In Eq. (1), P ijk (i =0, 1, …, n u , j =0, 1, …, n v , k =0, 1, …, n w ) are the control points, and Ni,pu(u), Nj,pv(v)and Nk,pw(w)are the univariate Bspline basis functions, constructed on the knot vectors U, V and W, and with degree p u , p v and p w , respectively. In our model, we assume that the u, v and w directions are parallel to the Cartesian axes previously introduced, and p u =p v =p w =2 (i.e., triquadratic B-splines are considered). After the generation of the trivariate B-spline solid and of the corresponding discretized volume with cubic voxels of 109 μ m size, the graphic visualization of the associated B-spline surfaces is exploited for adipose tissue segmentation, in combination with histogram analysis. The aim is to identify the grey-level threshold below which we assign the corresponding voxels to fat. Then, we extract the skin layer by means of a radial-geometry edge detection scheme applied to each coronal section with thickness equal to voxel size, deriving a mean breast skin thickness around 1.5 mm [26]. With this segmentation procedure, we prevent the inclusion of regions adjacent to the inner skin boundary with similar grey levels, like blood vessels. After the application of a morphological erosion operation to the skin layer, we assign the remaining voxels to fibroglandular tissue, including vasculature and possible muscle region. Then, we re-add the skin, varying the maximum number of voxels adjoined along radial directions, in order to obtain phantoms with different skin thickness. The pectoralis muscle is usually not included in dedicated BCT images and CC view mammograms, while a significant portion of it typically appears in MLO view mammograms. This requires modifying the obtained breast phantoms specifically for the MLO projection, by Fig. 2. (a) Schematics of the breast transversal (xy), sagittal (xz) and coronal (yz) planes, with the corresponding BCT slices and the main breast dimensions (i.e. the distance d from the torso to the nipple, the width w from the centre towards the sternum, and the upper part height h). (b) Schematic examples of the effects of the considered deformations when applied to the right breast, for different sets of transformation parameters. In some cases, the considered values are outside the realistic intervals suggested in Table 1, to provide a pictorial idea of the specific role of the parameters on the shape deformation. M. Oria et al. Computers in Biology and Medicine 198 (2025) 111121 3 properly incorporating the pectoralis muscle from the armpit down to the posterior nipple line. To this aim, a paraboloid with an ellipsoidal cross-section and axis parallel to the projection direction is superimposed onto the posterior region of the phantom. In the overlapping area, the original tissue is replaced with muscle tissue. To soften the transition at tissue interface, the muscle surface is slightly eroded, blurring the edges created by the Boolean operation. Additionally, the phantom is extended towards the sternum to accommodate the compression of a larger portion of the breast, including the muscle. This extension remains outside the compression paddles and does not appear in the image, serving solely to provide the necessary thickness for realistic breast compression. 2.2. Global deformation application Geometric transformations are applied to the reference phantom, replicating realistic deformities that can be observed in breast shape. The geometrical structure is altered by shifting voxels from their original positions to new positions without modifying the tissue assignment. The data are then interpolated on the input regular grid, with the nearestneighbour approach. Following Ref. [27] we consider: (i) turn deformation, occurring when the breast points towards the external flank; (ii) flatten-side deformation, describing the flattening of the breast towards the sternum; (iii) ptosis, which models sagging effects; (iv) top-shape deformation, which simulates variations in concavity of the breast top half; (v) turn-top deformation, occurring when the breast top half turns towards the shoulder. The transformations are expressed as a function of the main breast dimensions shown in Fig. 2a, i.e. the distance from the torso to the nipple for the x-axis direction (d), the width from the centre towards the sternum for the y-axis direction (w), and the upper part height or maximum length from the centre towards the clavicle for the z-axis direction (h). To obtain realistic deformations of the breast shape, the relevant parameters of each transformation should be approximately varied within the ranges listed in Table 1, where we have also indicated the interested plane and the views on which the specific transformation produces a visible effect. Turn deformation, which is responsible for the breast bending towards the body lateral part, can be described by scaling the lateral position (y) of a point versus its distance (x) from the torso, as: xʹ=x yʹ=y−w[a0(x d)+a1(x d)2] zʹ=z .(2) The turn towards the external flank can be modelled by assigning positive values to parameters a 0 and a 1 in the case of the right breast, while negative values have to be considered for the left breast. Realistic deformations can be obtained by varying the absolute values of a 0 and a 1 within the ranges 0−0.3 and 0−0.5, respectively, as reported in Table 1. Flatten-side deformation produces a compression of the internal part of the breast, transforming only the points with y >0 for the right breast, and with y <0 for the left breast. This can be described by modulating the x-coordinate of a point versus its lateral position y, as: xʹ=xf(y) = x[b3(y w)3 +b2(y w)2 +b1(y w)+b0] yʹ=y zʹ=z .(3) Parameters b i (i =0−3) can be derived by imposing the following constraints on function f(y): f(0) = 1,f(w) = f0 f’(0) = 0,f’(w) = f1,(4) where f 0 and f 1 control the flattening in correspondence of the sternum and breast middle part, respectively. It results: b3=f1+2−2f0,b2= − f1−3+3f0,b1=0,b0=1. Realistic deformations can be obtained by varying f 0 and f 1 within the ranges 1.5−5 and 0−f 0 3/2 , respectively (Table 1); no deformations are introduced when f 0 =1 and f 1 =0. Ptosis deformation, which causes breast drooping, can be simulated by modifying the vertical position (z) of a point as a function of its distance (x) from the torso, according to: xʹ=x yʹ=y zʹ=z−h[c0(x d)+c1(x d)2] .(5) Realistic deformations can be simulated by changing parameters c 0 and c 1 within the ranges 0−0.6 and 0−0.3, respectively (Table 1). Top-shape deformation, which produces a modification of the curvature of the upper part of the breast, can be described by scaling the zcoordinate of a point (if z >0) versus its x position, as: xʹ=x yʹ=y zʹ=zg(x) = z[e5(x d)5 +e4(x d)4 +e3(x d)3 +e2(x d)2 +e1 x d+e0] .(6) Parameters e i (i =0−5) can be derived by imposing the following constraints on function g(x): Table 1 List of breast geometrical transformations with the indication of the interested plane, the affected views and the interval of variations of the parameters leading to realistic deformations. Geometrical transformation Interested plane Affected views Parameter ranges Right breast Left breast Turn deformation xy CC 0 ≤a 0 ≤0.3 −0.3 ≤a 0 ≤0 0 ≤a 1 ≤0.5 −0.5 ≤a 1 ≤0 Flatten-side deformation xy y >0 (right) CC 1.5 ≤f 0 ≤5 1.5 ≤f 0 ≤5 y <0 (left) 0 ≤f 1 ≤f 0 3/2 0 ≤f 1 ≤f 0 3/2 Ptosis deformation xz MLO 0 ≤c 0 ≤0.6 0 ≤c 0 ≤0.6 0 ≤c 1 ≤0.3 0 ≤c 1 ≤0.3 Top-shape deformation xz (z >0) MLO 0 ≤s 0 ≤0.5 0 ≤s 0 ≤0.5 −2 ≤s 1 ≤ − 1−2 ≤s 1 ≤ − 1 0 ≤t 0 ≤20 0 ≤t 0 ≤20 2 ≤t 1 ≤15 2 ≤t 1 ≤15 Turn-top deformation yz (z >0) CC, MLO 0 ≤l 0 ≤0.3 −0.3 ≤l 0 ≤0 0 ≤l 1 ≤0.5 −0.5 ≤l 1 ≤0 M. Oria et al. Computers in Biology and Medicine 198 (2025) 111121 4 g(0) = 1,g(d) = 1 g’(0) = s0,g’(d) = s1 g’’(0) = t0,g’’(d) = t1 ,(7) where s 0 and s 1 are the slope parameters at the torso and nipple, respectively, while t 0 and t 1 are the relative curvature parameters. It results: e5= − 1 2t0−3s0−3s1+1 2t1,e4=3 2t0+8s0+7s1−t1, e3= − 3 2t0−6s0−4s1+1 2t1,e2=1 2t0,e1=s0,e0=1 .(8) Realistic deformations can be obtained by varying parameters s 0 , s 1 , t 0 and t 1 within the ranges reported in Table 1. Turn-top deformation, which modifies only the upper part of the breast (z >0) producing again a lateral bending as turn deformation, can be simulated by varying the y-coordinate of a point versus its vertical position z, as: xʹ=x yʹ=y−w[l0(z h)+l1(z h)2] zʹ=z .(9) To model realistic deformations, the absolute values of parameters l 0 and l 1 should be varied within the intervals 0−0.3 and 0−0.5, respectively (Table 1). Positive values have to be assigned in the case of the right breast, while negative values have to be considered for the left breast. To give a pictorial description of the effects of the considered deformations, Fig. 2b reports a list of images showing the modifications of a schematized section of the right breast in the plane involved in the transformation, for different selections of the parameter set. Values outside the realistic intervals suggested in Table 1 are also considered, to better highlight the geometric transformation effects. 2.3. Synthetic image generation For generating the synthetic mammograms from the 3D breast phantoms, we first compress the phantoms, after having added a few slices of muscle to the more internal coronal slice, by means of the finite element biomechanics solver FEBio integrated in VICTRE [28]. This enables us to model the compression phase, once specified the target thickness, which is defined on the basis of the percentage of fibroglandular tissue. To fix the breast compression thickness, we refer to the volumetric breast density grade scores proposed by Volpara® and correlated to the visual BI-RADS® 5th edition density categories [29, 30]. For the derivation of the mammograms in CC view, the compression is simulated by positioning the breast between two horizontal rigid plates parallel to the xy-plane, while for the MLO view the two parallel plates are rotated 45◦. The compressed phantoms are then used as an input to the Monte Carlo x-ray solver integrated in VICTRE, which enables us to obtain both the “for processing” and “for presentation” images, reproducing the image processing of the Siemens Mammomat Inspiration system. According to Ref. [21], the x-ray energy spectrum is sampled from a pre-computed probability distribution function corresponding to a 28 kVp tungsten anode source with a 50 μ m rhodium filter and a 1 mm beryllium window [31]. A 200 μ m thick amorphous selenium direct-conversion detector, with 85 μ m resolution, is simulated, including a 1 mm thick protective cover and a focused anti-scatter grid with parallel strips. The distance between the x-ray source and the detector is fixed to 65 cm. The calculation of the x-ray projection is performed by setting the mass density of fat, fibroglandular tissue, added muscle and skin plus nipple at 0.92, 1.035, 1.05 and 1.09 g/cm 3 , respectively; for masses we fix the density to 1.2 g/cm 3 , and for calcifications to 1.8 g/cm 3 [32,33]. It is important to point out that despite the produced mammograms contain 85 μ m pixels, these are acquired on phantoms with 109 μ m sized voxels, derived from BCT images with 273 μ m resolution. This means that the synthetic images inherit the low resolution of the source data, with the limitation that fine-scale features cannot be well depicted. 3. Results and discussion In the following, the methodology described in Section 2is applied to generate a sample of synthetic mammograms, with different breast shape, size, tissue composition, and type of lesion. To this aim, we consider four sets of dedicated BCT images acquired on four patients, which enable us to produce four reference phantoms, independent from each other in terms of tissue composition. 3.1. Reference phantom generation and synthetic mammogram acquisition The reference phantom construction is depicted in Fig. 3, for a stack of dedicated BCT images acquired on a specific patient, here named patient #1. By combining the BCT slices on the coronal view, we obtain a 3D grid of 624 ×592 ×243 pixels, with size of 273 μ m. A trivariate triquadratic B-spline solid is then built using the 3D grid to produce a digital phantom, composed of 109 μ m cubic voxels with associated grey levels. The tissue classification is performed by analysing the histogram of the overall grey levels obtained after B-spline approximation. This is shown in Fig. 3a, together with the Gaussian fit, the residual histogram, derived by subtracting the Gaussian fitting curve, and the Gaussian fit of the residual. From the comparison of the two peaks amplitude and width, it is possible to determine the tissue composition, considering that the interval in Hounsfield units for fat typically ranges from −200 to −50 HU [34,35]. For the specific breast here examined, the peak with the largest amplitude and width, corresponding to fat, is located at −173 HU, and we define a threshold of −135 HU to separate it from fibroglandular tissue; then, we identify the skin layer. The tissue distribution calculated slice-by-slice along the x-direction, from the thorax muscle to the nipple, is reported in Fig. 3b. Globally, when the full skin layer is considered, the phantom volume is composed of 85 % fat, 11 % fibroglandular tissue and 4 % skin (including the nipple region). Within the category of fibroglandular tissue we also include vasculature and muscle, having similar mass density and thus comparable x-ray attenuation for synthetic mammogram generation. The top, side and front views of the obtained right breast phantom, used as a reference for the transformations reported in Sub-sections 3.2 and 3.3, are depicted in Fig. 3c, together with a sagittal section representing the tissue internal distribution, corresponding to a heterogeneously dense breast (Fig. 3d). The main phantom dimensions, as defined in Sub-section 2.2, result to be d =9.5 cm, w =5.3 cm and h =5.6 cm. According to the volumetric breast density grade scores proposed by Volpara® and considering the specific percentage in volume of fibroglandular tissue (11 %) found in the constructed phantom, we opt for a breast compression thickness of 6 cm for the calculation of the x-ray projections [36,37]. The synthetic mammograms, generated in the “for presentation” form, are reported in Fig. 4a for the CC view and in Fig. 4b for the MLO view. In order to simulate skin loss, which can occur in real digital mammograms, due to image saturation above a specific x-ray exposure [38], the images are derived for different skin thicknesses, from less (0.109 mm) to more (1.090 mm) realistic values. It is important to note that, in addition to the projection related to the different tissue attenuation coefficients, the derivation of “for presentation” images in VICTRE pipeline requires the calculation of the projection of a homogeneous phantom, with the same shape and voxel decomposition of the original 3D breast phantom but entirely made of adipose tissue. In such a way, it is possible to obtain a map that allows us to identify the regions of the breast with reduced thickness along the M. Oria et al. Computers in Biology and Medicine 198 (2025) 111121 5 compression direction, like the skin. Then, the “for presentation” image is calculated by multiplying the grey level at each pixel in the original projection by the inverse of the greyscale value of the corresponding pixel in the homogeneous phantom projection. This process enables us to clearly visualize the total skin edge, which appears as a thick light grey line that follows the entire breast boundary, when the input skin thickness is within the normal range, i.e. 0.7−2.3 mm [39]. To obtain mammograms more similar to the real ones, where the skin is not so evident and resolution artefacts are present at the breast boundary, we opt for a decrease in the skin thickness down to 0.545 mm, reducing the number of skin voxel layers to 5. This value is selected for the following modelling analysis. To demonstrate the application of the methodology in different cases, Fig. 5 shows the results obtained on other three sets of BCT images acquired on three different patients, illustrating, on the left, the original BCT images compacted in 3D objects, in the centre, the percentages of main tissues per each coronal slice from muscle to nipple, extracted from trivariate B-spline solids, together with the reconstructed reference phantoms, and, on the right, the calculated mammograms in CC view. The derived phantoms differ in size, being about 11.4 ×12.7 ×15.1 cm 3 for patient #2, 8 ×9.2 ×11.5 cm 3 for patient #3, and 10 ×12 ×14.8 cm 3 for patient #4. These have a variable percentage of fibroglandular tissue, whose segmentation is performed after fixing the thresholds below which we assume to have fat. After histogram analysis, the selected values are −115 HU, −45 HU and −115 HU, for the phantoms of patients #2, #3 and #4, respectively. Following this choice, phantoms from patients #2 and #3 contain a percentage of fibroglandular tissue of 17 % and 21 %, respectively, thus according to Volpara® volumetric density grade scores they belong to the class of extremely dense breasts; for them, we opt for a compression of 5 cm. Phantom from patient #4 is composed of 15 % fibroglandular tissue and can be associated with the class of heterogeneously dense breasts; this is compressed to 5.5 cm. The generated mammograms reflect the variability in tissue composition, with the ones acquired on phantoms #2 and #3 characterized by high density areas distributed throughout the breast shape, and the fourth one interested by a more compacted fibroglandular region. 3.2. Application of geometrical transformations Starting from the reference phantoms previously reconstructed (see Figs. 3 and 5), we can generate multiple synthetic mammograms in both CC and MLO views, by simply applying the geometrical transformations described in Sub-section 2.2. To obtain realistic deformations of the breast shape, the relevant parameters of each transformation should be approximately varied within the ranges listed in Table 1. It is clear that by properly setting the transformation parameters, it is possible to obtain a large number of digital phantoms and thus synthetic mammograms. Some examples are shown in Fig. 6 for the reference breast phantom #1, distinguishing between the transformations that produce shape variations mainly visible in a specific view (CC or MLO) and in both. The shape modifications can be appreciated by comparing the deformed phantoms depicted on the top of Fig. 6 to the transversal, sagittal and coronal views of phantom #1 in Fig. 3c. Focusing on the transformations responsible for breast shape modifications in the transversal planes and thus observable in CC view, the images shown in Fig. 6a are obtained for turn deformation by setting parameters a 0 and a 1 at 0.001 and 0.47, and for flatten-side deformation by fixing f 0 =2.5 and f 1 =4.5. By comparing these mammograms to the ones in Fig. 4a, it is well evident how the turn deformation is responsible for a more pronounced breast curvature at the external side, which can be incremented by increasing parameter a 0 . On the contrary, the flattenFig. 3. (a) Histogram of the overall grey levels obtained after B-spline approximation of the set of dedicated BCT images from patient #1, used for the choice of the threshold for the classification of fat and fibroglandular tissues. The graph also contains the relative Gaussian fit, the residual histogram, derived by subtracting the Gaussian fit curve, and the Gaussian fit of the residual. (b) Percentages of fat, fibroglandular and skin tissues per each coronal slice from muscle to nipple. (c) Transversal (left), sagittal (centre) and coronal (right) views of the obtained reference breast phantom. (d) Phantom section on median sagittal plane (xz), with the spatial distribution of the different considered tissues. M. Oria et al. Computers in Biology and Medicine 198 (2025) 111121 6 side causes a compression of the internal part, which can be amplified for higher values of f 0 . Considering the transformations that lead to sagittal plane variations and thus detectable in MLO view, the images in Fig. 6b are derived by setting c 0 =0.6 and c 1 =0.15 for ptosis, and s 0 =0.1, s 1 = − 2, t 0 =20 and t 1 =3 for top-shape deformation. In the first case, a strong modification of the breast profile is produced with respect to the images shown in Fig. 4b, with a noticeable hanging down effect, which can be further raised by increasing c 1 . In the second case a compression of the top part is obtained, with a curvature variation, which can be made more pronounced for higher values of t 1 . Finally, the effects of turn-top deformation, which contributes to modifications in the coronal planes, are depicted in Fig. 6c, for both CC and MLO images, obtained with l 0 =0.175 and l 1 =0.2. Even if this deformation produces a significant deflection of the top-half of the breast towards the external side, the obtained mammograms result to be very similar to the reference phantom ones, due to the negligible profile variations on the projection planes, for both CC and MLO views. Additional phantoms and relative synthetic mammograms can be obtained by properly combining in sequence the above geometrical transformations and carefully tuning their parameters. 3.3. Insertion of lesions Another way to expand the range of images is the artificial insertion of lesions within the reference digital phantoms, by means of a replacement of tissue at voxel level. In Fig. 7, we have reported the CC view images calculated after the inclusion of realistic lesions, which can be observed in mammograms with anomalies. As an example, Fig. 7a illustrates how the image of the reference phantom #1 appears after the introduction, in the lower-outer quadrant, of a mass with well-defined margins. This mass, which has a volume of 220 mm 3 and an average diameter of 7.2 mm, can be representative of a cyst, a lump, or in general a benign lesion. The case of a malignant lesion is illustrated in Fig. 7b, with the appearance of a spiculated region characterized by sharp lines radiating from its margin. This is created by inserting again in the lowerouter quadrant a 4 mm sized mass with spicules with length varying from 1 to 6 mm. Another example of anomaly is represented by calcification clusters, like the one simulated in Fig. 7c, where a group of small white spots clearly emerge in the image, due to the very high density of these localized regions with respect to the surrounding tissues. In particular, four lobular calcifications are inserted in the upper-outer quadrant, with volume varying from 1.7 to 3.4 mm 3 . The digital model of the different types of lesions can be generated by exploiting the VICTRE pipeline, which includes both a breast mass generation software [40] and a calcification cluster generation algorithm. This approach has been followed for the creation of the lesions shown in Fig. 7a, b and 7c. Alternatively, it is possible to use the MaXIMA Breast Lesions Models Database, which provides lesion models with realistic shape, obtained from segmentation of CT and tomosynthesis images as well as from mathematical algorithms [41,42]. By varying size, shape and position of the lesions, several images can be generated, contributing to the population of datasets with different pathology subtypes. Tubular calcifications appearing as linear or rod-like calcium deposits [43] can also be added to the phantom after a proper segmentation of its blood vessels and ducts. To facilitate the segmentation process, we apply the multiscale vessel enhancement filter, developed by Frangi et al. [44], to the trivariate B-spline solid introduced in Sub-section 2.1, analysing its multiscale second-order local structure (Hessian matrix). Using the Vascular Modelling Toolkit (VMTK) software [45,46], a tool widely adopted for the geometric analysis and Fig. 4. Visualization of the biomechanical compression of the reference breast phantom from patient #1 and synthetic mammograms generated with skin thickness variable from 0.109 to 1.090 mm, for (a) craniocaudal (CC) and (b) mediolateral oblique (MLO) views. M. Oria et al. Computers in Biology and Medicine 198 (2025) 111121 7 reconstruction of tubular structures in medical images and in silico testing [47], we define a vesselness 3D function at different scales s, positive outside the vessel-like structure and negative inside it, namely: V(s) = [1−exp(−R2 1 2 α 2)]exp(−R2 2 2β2)[1−exp(−S2 2 χ 2)].(10) In (10) R 1 , R 2 and S are three measures, which are introduced to distinguish between plateand line-like structures (R 1 ), to discriminate blob-like structures (R 2 ), and to quantify grey-level contrast (S), for differentiating the background from the relevant structures. Parameters α , β and χ are thresholds that control the sensitivity of the filter to the measures R 1 , R 2 and S, respectively; here, we select α =0.5, β =0.5 and χ =7. We also define the minimum (s min ) and maximum (s max ) scales of the structures of interest, which can be chosen so that they cover the entire range of vessel and duct widths; here, we fix s min to 0.7 and s max to 2. Then, we integrate the vesselness measure provided by the filter response at different scales between s min and s max to obtain a probabilitylike estimate of vesselness. To introduce tubular calcifications in a specific breast region, we assign to all the phantom voxels belonging to this region and with vesselness value within a specified interval the material properties of calcification. As an example, Fig. 7d shows a mammogram where vascular calcifications are inserted in the upper-outer and lower-outer quadrants, in proximity to the skin, considering a range of vesselness from −0.4 to −0.1. With this choice, the introduced calcifications do not replace the entire vessel volume, guaranteeing the presence of a lumen inside. By tuning the extension of the vesselness interval, we can vary the calcification status. 3.4. Quality assessment of the synthesized mammograms at texture level The quality of the synthesized mammograms is evaluated in comparison to real ones by calculating a series of metrics, which can provide a quantitative measure of texture similarity. The analysis is performed on regions of interest (ROIs) with 300 ×300 px size, mainly containing a specific type of tissue (adipose or fibroglandular). The real ROIs are extracted from mammograms acquired with Siemens scanners, belonging to the Vietnamese dataset of digital mammography (VinDrMammo) [48–50]. We estimate the self-similarity of texture at different scales by means of the fractal dimension (FD), derived from both box-counting method Fig. 5. Visualization of the results obtained from BCT images acquired on patients #2, #3 and #4. On the left: original BCT images compacted in 3D objects. In the centre: percentages of main tissues per each coronal slice from muscle to nipple, and reconstructed reference phantoms in coronal view. On the right: synthetic mammograms calculated in CC view. M. Oria et al. Computers in Biology and Medicine 198 (2025) 111121 8 and FFT-based power spectrum analysis, for the latter FD = (8−β)/2, where β is the average log-log slope [51–54]. In parallel, we evaluate the entropy within the extracted ROIs, to measure the texture complexity by quantifying its degree of disorder/uniformity [55]. For the above metrics, greater texture similarity between synthetic and real ROIs is indicated by closer values. Moreover, the synthetic ROIs are directly compared with those extracted from real mammograms by calculating the Feature Similarity (FSIM) index, which enables us to measure similarity by means of phase congruency and gradient magnitude feature maps [56]. It ranges from 0 to 1, where 1 indicates perfect similarity. To provide a quantitative comparison with well-established in silico approaches, the analysis is also performed on ROIs extracted from mammograms generated by using the entire VICTRE pipeline, including phantom creation with a voxel resolution of 50 μ m, thus compatible with the resolution of mammography. All the metric values are reported in Table 2, considering two sets of ROIs with a prevalence of fibroglandular or adipose tissue. For both the two compared methods for generating synthetic images (i.e. our proposed approach using patient–based phantoms and the one exploiting the entire VICTRE pipeline), the greatest similarity to the real ROI is achieved for the fibroglandular tissue ROIs. For the case of adipose tissue, better scores are obtained for the ROI generated with the patient–based phantom approach, with a higher value of FSIM, and values of FDs and entropy closer to the ones evaluated for the real ROI. Finally, a short visual grading analysis (VGA) is performed by the two co-authors from the Dutch Expert Centre for Screening (LRCB) in the Netherlands, both experts in radiology with a focus on mammography. The image quality criteria adopted for the assessment of the realism of the synthesized mammograms, and the relative scores, are summarized in Table 3, distinguishing among anatomical structures and tissues. They point out that the spatial distribution of fibroglandular and adipose tissues throughout all the breast shape is sufficiently realistic, the greyscale of fibroglandular tissue and soft tissue lesions are well reproduced, and the skin line and the nipple are adequately constructed. However, the obtained images are easily distinguishable as synthetic, due to the low resolution inherited from the BCT images. As a consequence, the Cooper’s ligaments are not clearly visible, the vessels contain gaps and cannot be seen in dense areas, and the ducts are not sharply depicted. Also the representation of soft tissue lesion margins and calcifications suffers from the low resolution of the images, needing to be improved. Fig. 6. Examples of breast phantom deformations (top) with associated synthetic mammograms (bottom): (a) turn and flatten-side deformations with relative CC images; (b) ptosis and top-shape deformations with relative MLO images; (c) turn-top deformation with relative CC and MLO images. The geometric transformations are applied to the reference breast phantom from patient #1. Fig. 7. Examples of synthetic mammograms in CC view generated from the reference phantom from patient #1 by including artificial lesions of different type: (a) round mass with well-defined margins; (b) spiculated mass; (c) cluster of calcifications; (d) vascular calcifications. M. Oria et al. Computers in Biology and Medicine 198 (2025) 111121 9