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Algorithmic 3D simulation of breast calcifications for digital mammography
Computer Methods and Programs in Biomedicine 66 (2001) 115– 124 www.elsevier.com/locate/cmpb
Algorithmic 3D simulation of breast calcifications for digital mammography Janne Na¨ppi a,*, Peter B. Dean b, Olli Nevalainen a, Sakari Toikkanen c a
Department of Computer Science, Uni6ersity of Turku, Turku Centre for Computer Science (TUCS), Lemmink aisenkatu 14 A FIN-20520, Turku, Finland b Department of Diagnostic Radiology, Uni6ersity of Turku, Medical Imaging Center, Uni6ersity Central Hospital, FIN-20520, Turku, Finland c Department of Pathology, Uni6ersity Central Hospital, FIN-20520, Turku, Finland
Abstract We present a framework for algorithmic three-dimensional simulation of breast calcifications. The simulated calcifications can be viewed from any angle at a higher spatial resolution than currently available for digital mammography, and they can be placed onto a simulated or real mammographic background to provide example cases for computers and radiologists. In order to simulate calcification clusters, we also show how to simulate duct networks and terminal ductal lobular units. We evaluated the model with a double-blind evaluation of 60 cases with four experienced radiologists by mixing 30 cases of simulated calcification clusters on a real or simulated mammographic background with 30 cases of real breast calcification clusters digitized at a spatial resolution of 15 mm from high-resolution radiographs of 5 mm slices of breast specimens. The results indicate that the majority of the 2D projections of the 3D simulated calcifications compare favorably with the radiographic images of real breast calcifications. © 2001 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Breast calcifications; Simulation; Computer-assisted diagnosis; Digital mammography
1. Introduction Fewer than 20% of the cases with clustered breast calcifications are produced by malignant diseases. To identify the suspicious cases and to prevent unnecessary interventional procedures, a detailed and reliable analysis of the mammographic appearance of the calcifications is required [1–3]. * Corresponding author. E-mail address: [email protected] (J. Na¨ppi).
Computer-assisted diagnosis (CAD) [4] may be used to improve the speed and accuracy of the analysis of calcifications, but the enormous number of variations in the appearance of the calcifications on a mammogram poses a serious challenge. Ideally, the design and evaluation of the computer algorithms should be based upon carefully documented cases in which the calcifications seen on histopathologic examination are the same as those demonstrated mammographically. Since arranging this kind of documentation is extremely time-consuming, most mammographic
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images of breast calcifications are fairly imperfect representations of the actual pathological entities, and the limited spatial resolution of digital images may introduce further degradation. Furthermore, the continuous improvement in the quality of digital mammograms makes current digital archives of mammographic cases soon outdated, so they cannot be considered as a fully reliable reference material for the important task of developing and evaluating CAD. Simulation models that generate synthetic images of lesions in clinically relevant detail provide an alternative source for mammographic images. A particular advantage of the simulation models is that the image formation is fully controlled and exactly reproducible. The previous computer models for simulating breast calcifications were limited to 2D [5–8], and an inherent problem of these models was that they tried to reproduce the 2D-projections of objects that are actually 3D entities. In contrast, the 3D models presented in this paper can be based directly on the knowledge of anatomical features and the pathological processes through which real breast calcifications are produced. Other work on mammographic 3D simulation has only considered the simulation of complete breast at a low spatial resolution [9– 11]. As far as we know, our work on the algorithmic 3D simulation of breast calcifications has been unique. The major parts of our model are: (1) simulation of the X-ray image acquisition (Section 2.1); (2) algorithmic 3D simulation of the calcification particles of specified type (Section 2.2); (3) 3D simulation of the breast structures associated with the calcifications (Section 2.3); and (4) placing of the calcifications onto a real or simulated mammographic background (Section 3). The evaluation of the simulation model is presented in Section 4, with discussion and conclusions in Sections 5 and 6. The simulated calcifications can be applied to the training and evaluation of CAD algorithms and radiologists, but an analysis of these applications is beyond the scope of this paper and will be considered elsewhere. The software associated with the model has been implemented using Khoros [12].
2. The simulation model
2.1. X-ray image acquisition The X-ray image acquisition model is based on the principles of the actual physical process [13]. There are three components: an X-ray field, a simulated breast 6olume, and a film plane (Fig. 1). For each pixel in the film plane, the X-rays are emitted from the X-ray source through the simulated breast volume onto the film plane. The X-ray field is represented as an image pair (E( ,P( ), where the energy field E( specifies the current X-ray energy and the position field P( specifies the current distance to the film plane. Each pixel of E( is initialized to the X-ray energy for which the maximal transferred X-ray energy at the film plane will be 1 unit, and each pixel of P( is initialized to the distance between the X-ray source and the film plane. The simulated volume may contain a calcification surrounded by homogeneous breast tissue, a breast structure (i.e. duct network or a terminal ductal lobular unit) with calcifications, or a complete simulated breast with breast parenchyma, fatty tissue, breast structures, and calcifications (Fig. 2). The value of each voxel within the volume specifies the linear attenuation coefficient (LAC), i.e. the relative radiographic density of the matter at that position. For a homogeneous object with LAC m and thickness x, the amount of X-ray attenuation can be computed using Beer’s law of photon absorption [13]:
Fig. 1. The X-ray image acquisition model. The X-ray field is represented as an energy-position image pair (E( ,P( ).
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Fig. 2. The breast model is composed of independent modules, each of which may be imaged separately.
N =N0e − vx
(1)
Here N0 is the number of the incident X-ray photons and N the number of transmitted X-ray photons. Normal biological variation is simulated by allowing a small random variation of the LAC. The digital images of the model are represented nominally at a spatial resolution of 10 mm with 12-bit pixels. Specification of an explicit nominal resolution is not really necessary, but it greatly simplifies the application of the simulation model. However, although the high spatial resolution produces realistic results, the density modeling and computation of the X-ray penetration quickly become a computational burden as we increase the size of the simulated volume. Therefore we minimize the use of volumes by approximating the X-ray penetration mathematically in the space between the simulated volumes using Eq. (1) and the geometric relations of the breast model: the X-ray energy E(m, n) is decreased according to the LAC of the material through which the X-ray currently traverses, and the position value P(m, n) is decreased according to the traversed distance. At the film plane we have P(m, n) =0 for all positions (m, n). The film response D(m, n) is modeled by ([14], p.273) D(m, n)=k log10(− (m, n)) − d0
(2)
where g is the gamma of the film, (m, n) is the incident photon intensity at (m, n), and d0 is an optical intensity constant. The effects of film noise, scattering and other image noise are modeled simply by convolving the final X-ray energy field with a 2D Gaussian
G(m, n)=
1
2y| 2
− (m2 + n2)
e
2|2 −
(3)
where s controls the magnitude of the noise. The effect of digitization is simulated by computing the local mean of the image and downsampling it to the final spatial resolution using bilinear interpolation.
2.2. Calcification models The models of calcifications and breast structures are based on their anatomic properties that have been described in the literature [1], and observed in practice using preoperative mammograms, specimen radiographs, stereomicroscopy, and histopathologic images. We have collected this knowledge in our experimental differential diagnosis system which is similar to (but more detailed than) the American College of Radiology Breast Imaging Reporting and Data System (BIRADS). We have developed three principal models for modeling calcification particles: irregular, o6oid, and elongated. The other types of calcifications can be derived from these basic types with appropriate parameter setting and application of the associated breast structures. The calcifications can be viewed from a specified direction by rotating the calcifications in 3D ([15], pp. 223–224) before projecting the X-rays through the 3D LAC volume modeling the calcification. Irregular particles are simulated using a 3D random-walk model ([16], pp. 232–255). The model is controlled by two parameters: shape asymmetry [0,1] and boundary roughness
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[0.1]. First all voxels within the target volume V are initialized to the LAC of fatty tissue. Next the calcification is constructed by starting N()[5, 20] (N() =5 + 15) random-walk processes from the center of V. Each process assigns the LAC of calcium to the value of the current voxel, and then moves randomly to one of the neighboring voxels. Each process continues until it reaches the boundary of the surrounding volume, or until it has assigned V P() (P()[0.01, 0.125]) voxels to the calcification. At this stage the particles are rather sparse cloudy objects. Indeed, such particles can be used to simulate powderish clusters. To produce standard solid particles, we apply D() [2, 10] morphological dilations to the calcification using a spherical morphological element. Thus the model simulates the apparently irregular growth of the malignant types of calcifications. It is also simple, fast, and produces realistic results. Ovoid types of calcifications usually develop within lobules. The simulation of these mostly round particles is based on modifying the shape of an ellipsoid. The simplest case is the ovoid solid type (pearl), which is implemented by filling the ellipsoid with the LAC of calcium. Shape deformation is introduced by varying the axis ratio of the ellipsoid. Eggshell type is computed similarly, except that now only the surface region of specified thickness represents the calcium while the density of the center region is set to a LAC between that of calcium and fatty tissue. Milk-of-
calcium type is derived from the eggshell model by removing the top of the eggshell with a large sphere. Some powderish ‘sediment’ of high density is added randomly to the bottom region of the teacup. Simulation of elongated types is based on a wireframe model which is similar to the duct models discussed in Section 2.3. The wireframe defines the overall shape of the calcification, with the calcification residing within a specified radius from the wireframe. Various types of branching shapes can be generated similarly. For example, the plasma cell mastitis types are mostly quite straight and smooth, while the (solid) casting types are more twisting. The intraductal case is implemented as a solid wireframe, and the periductal case is implemented as a hollow wireframe. Examples of the simulated irregular, ovoid, and elongated particles are seen in Fig. 3. Some of the simulated calcifications may look overly regular, and although such calcifications appear in real cases (in high-quality images), it is sometimes necessary to simulate calcifications that look less than perfect. Then we apply a degradation algorithm to improve the realism of the simulated calcifications. The algorithm squeezes the calcification with a plane perpendicular to the origin by moving the intersected calcification voxels to the next plane closer to the origin (Fig. 4). The process roughly simulates the internal tissue forces that limit the growth of the calcification. The parameter 1 [0, 1] characterizes the maxi-
Fig. 3. Examples of simulated irregular (top), ovoid (middle), and elongated (bottom) particles.
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Fig. 4. The first steps of the degradation algorithm (top): the particle is squeezed by a plane, rotated to a random orientation, and squeezed again. Bottom row: a perfectly round calcification (left), and results of the degradation algorithm.
mum relative distance which the squeezing plane can move from the surface towards the origin. The squeezing is repeated for o(2) [9, 20] (2 [0, 1]) random orientations. Despite its simplicity, the algorithm produces visually good results.
2.3. Models for breast structures and mammographic background The growth pattern of most calcification clusters relates to the location and shape of the breast structures within which the calcification particles appear. The most important breast structures in this respect are the duct network and the terminal ductal lobular units (TDLUs). Modelling of these structures is necessary for getting a detailed simulation of 3D calcification clusters. Visualization of biological ducts ([17,18], p. 30; 3, pp. A4 – A6) shows tree-like patterns which can be represented compactly by fractal geometry [10,11,19]. In most cases we use a random midpoint displacement algorithm [16]. The algorithm is initially applied to a two-point (poly)line P0 = [(x00, y00, z00), (x01, y01, z01)] which roughly defines the ultimate length and orientation of the final duct network. The algorithm then splits P0 at midpoint to produce a new polyline P1 =F(P0) = [(x10, y10, z10), (x11, y11, z11), (x12, y12, z12)]. By repeating this procedure to the prespecified precision we obtain a polyline L0 =Pn =F(Pn -1), which is then expanded to the next level by allowing some of the points (xni, yni, zni) to branch in up to three new ducts (i.e. polylines). The al-
gorithm is applied recursively to each of the new ducts, until the duct system has been expanded to the specified number of levels (Fig. 5). The final duct network consists of a set L= (Li ) of connected polylines. The general form of L is controlled by the limiting bounds of the branching angle and the length of the polyline segments. The generation of the calcifications within the duct is controlled by associating attributes with the nodes of the polyline. A particularly useful attribute is the duct diameter, which for example allows simulation of the cases where cancer seems to spread from the ‘buds’ towards the main duct in the form of malignant calcifications. Ramification matrices are a useful alternative for generating the duct network [11,20]. In short, the matrix specifies the branching probabilities that generate the topology of a binary tree system. We can control the overall shape of the resulting duct system by introducing length, diameter, and deviation attributes which depend on the order of the nodes of the tree [20]. The flexible control over the final form makes ramification matrices especially useful for simulating the duct network of a complete breast, although several matrices and attribute functions are needed for truly realistic results. It is not necessary to place the simulated duct network into an actual volume, since we can place the calcifications simply by tracing the polyline representation and constructing the appropriate wireframe models for generating the individual calcification particles.
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Fig. 5. Examples of simulated duct networks. Top left: demonstration of the midpoint replacement algorithm. Bottom left: examples of the resulting 3D duct networks. Right: a duct network generated by two ramification matrices of a random binary tree [11, 20].
A TDLU is a terminal part of the duct system which consists of a large number of lobules, making it resemble a bunch of grapes. The calcifications develop within the lobules, and we only need to know the lobule coordinates in order to place the calcifications. To generate a TDLU, we model the lobules as spheres which are placed around the origin in a series of layers (Fig. 6), in which the distance between the neighboring spheres is minimized. The complete breast simulation models reported so far, including our own, cannot yet produce a truly realistic variable mammographic background. However, in most cases the simulation of calcifications only requires relatively small background regions, so we are currently using two simplified sources for mammographic background: real mammographic images and fractal texture. The fractal texture generator that we use produces visually realistic mammographic background. It is based on a recursi6e 2D midpoint displacement algorithm [16], in which at each step the current image is interpreted as a square lattice of pixels. After adding a new pixel to the midpoint of every lattice square, the image is rotated by 45 degrees to obtain a new image with a higher spatial resolution. The value of each new pixel is computed using the average of the surrounding pixels, the specified standard deviation, and the specified Hurst coefficient H [0,1]. Visual com-
parison between the fractal texture and real mammographic texture suggests that the texture generated with Hurst coefficients 0.1B HB 0.5 is most realistic (Fig. 7). For H\ 0.5 the image ‘noise’ becomes overly minimal. The fractal texture generator can only produce background that seemingly combines fatty tissue and breast parenchyma. Other distinctive breast features, such as ducts, Cooper’s ligaments, or tumors, should be modeled separately.
3. Placing the calcifications onto mammographic background To place the calcifications, it is usually convenient to use a contrast parameter Ci for character-
Fig. 6. A large TDLU being generated (left), and a small TDLU in which the lobule positions have been randomly varied. The spheres represent the maximal space available for each lobule.
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Fig. 7. Examples of simulated mammographic background. From left to right: H =0.20, H =0.33, H = 0.50, and H =0.75.
izing the level of embedding for each calcification i. For realistic results the contrast should vary slightly between each calcification of a cluster. A basic method is to compute the mean of the calcification (I mean ) and its target background c (I mean bg ),and then scale the intensities of the calcification to satisfy (I mean ) = I mean +I mean c bg bg Ci. Here we could also use maximum intensities and/or standard deviations. Other techniques include adjustment of the intensities in concentric rings based on a statistical analysis of a large number of real cases [5], and scaling of the image intensities according to the relative intensities of calcium and fatty tissue deduced from the background. All these techniques are applicable but produce somewhat different results. The 3D simulated calcifications can be placed onto mammographic background using these methods. However, the methods were originally developed for use with preoperative mammograms, and in specimen radiographs the results are less satisfactory because of the improved image quality and the large size variation of the calcifications. For example, some of the methods only modify the boundary region of the calcification, and the inconsistency between the center region and the background may be clearly visible. Here we use a novel method for combining 3D simulated calcifications with mammographic background. The idea is to interpret the mammographic background as a representation of the relative densities of the original mammographic
volume. First we simulate the X-ray penetration through the 3D calcifications, and then add the attenuation caused by the LACs derived from the background. The contrast of the calcification is controlled by varying the relative magnitude of the attenuation. Since the calcifications may nevertheless look too smooth and regular compared with the background, we can add random variation to the X-ray field coefficients relative to the standard deviation of the background intensities. Examples of placing a calcification onto mammographic background are seen in Fig. 8.
Fig. 8. Top: ideal calcification (left) is degraded (middle), and mammographic noise is added (right). Middle: simulated background before (left) and after placing the calcification (middle and right). Bottom: example results of our new method.
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Table 1 Results of four radiologists’ evaluation of 30 simulated and 30 real breast calcifications Summary of the radiologists’ evaluation of the calcifications
Correctly evaluated
Incorrectly evaluated
Undecided
Real calcifications (n =30) Simulated calcifications (n= 30) All calcifications (n = 60)
62% 54% 58%
19% 22% 20%
19% 24% 22%
4. Results We evaluated the simulation model by digitizing 50 specimen radiographs of thin (5 mm) breast specimens at a spatial resolution of 15 mm with a 10-bit intensity resolution, and extracting thirty 512*512-regions of interest containing calcifications. The extracted regions could contain both benign and malignant calcifications. Next we generated 30 cases of 3D simulated calcification clusters, and placed them onto both real (15 cases) and simulated (15 cases) mammographic background. The cases of real mammographic background did not contain any real calcifications. The original 10-bit images were scaled to 8-bit images to present them on a standard 17’’ PC computer screen. We performed the evaluation by showing randomly mixed real and simulated regions of interest with calcifications, one at a time, to four radiologists with considerable (10– 25 years) experience in interpreting mammograms. A questionnaire containing the following statement was presented to each radiologist: Some of these images contain real calcification(s) from specimen radiographs. The other images contain computer simulated calcification(s). Please evaluate each image according to whether the calcifications appear real or simulated. 0) No calcifications 1) Definitely real calcification(s) 2) Probably real calcification(s) 3) Could be either 4) Probably simulated calcification(s) 5) Definitely simulated calcification(s) The four radiologists differed in their evaluation according to their experience with the speci-
men radiographs on which the evaluation material was based, those with more experience having higher accuracy. As can be seen from Table 1, the real calcifications were only slightly more accurately evaluated than were the simulated calcifications. There was a greater interobserver variation with the simulated calcifications. The presence of a real or simulated mammographic tissue background had no effect upon the evaluation. Eight of the cases of simulated calcifications were correctly considered to be simulated by all of the radiologists, and in four of the cases of simulated calcifications none of the radiologists recognized them as simulated. A majority (18/30) of the simulated calcifications was considered to be real by at least one of the radiologists (Examples, Fig. 9).
5. Discussion The results of the radiologists’ evaluation indicate that some, but not all, of the examples of simulated calcification clusters do indeed appear as real calcifications. The results will help us to select those parameters that produce calcifications most reliably resembling true breast calcifications. For example, the preliminary analysis suggests that the realism of the simulated calcification clusters could be generally improved by increasing the size, shape and density variance of the particles, and by generating larger clusters. The evaluation was particularly demanding as the simulated calcifications were compared with very high quality, high-resolution (15 mm) digitized images of thin (5 mm) breast slices.
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6. Conclusion Algorithmically 3D simulated calcifications are an alternative to real mammographic data to produce cases for evaluating image processing algorithms and for teaching computers and radiologists. Images generated by 3D models are conceptually superior to those generated by 2D models: anatomic knowledge is readily applicable, the image formation is fully controlled, and the simulated calcifications can be placed onto any mammographic background. The results suggest that the more successful 2D projections of the 3D simulated calcifications compare favorably with high resolution (15 mm) specimen radiographic images of real breast cal-
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cifications. The more successful parameters are undergoing further development. Future work on the algorithmically 3D simulated calcifications involves development of a simpler user interface and further adjustment of the models, as well as their application in computerassisted diagnosis. A new evaluation study with a larger number of cases and a larger amount of highly realistic simulated cases than that used here is also planned.
Acknowledgements This work was supported by grants from the Instrumentarium Science Foundation, EVO Spe-
Fig. 9. Four distinct examples of the mammographic clusters of the evaluation image set. Top row: two real cases, labeled as real (left) and simulated (right) by all radiologists. Bottom row: two simulated cases, labeled as real (left) and simulated (right) by all radiologists.
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cial Funds, Turku University Foundation, and by the Finnish Academy. We would also like to thank the radiologists who participated in the evaluation.
[10]
[11]
References [1] L. Taba´ r, P.B. Dean, Teaching Atlas of Mammography, Thieme, Stuttgart, 1985. [2] M. Lanyi, Diagnosis and Differential Diagnosis of Breast Calcifications, Springer-Verlag, Berlin, 1986. [3] L. Taba´ r, Diagnosis and In-Depth Differential Diagnosis of Breast Cancer, EAR Teaching Programmes, Turku, Finland, 1996. [4] F. Shtern, M.W. Vannier, S.M. Pizer, D. Winfield (editors), Report of the Working Group on Digital Mammography: Computer-Aided Diagnosis and 3D Image Analysis and Display, Academic Radiology 6 (Suppl. 5) (1999) S231-S237. [5] F. Lefebvre, H. Benali, R. Gilles, R. Di Paola, A simulation model of clustered breast calcifications, Medical Physics 21 (1994) 1865 –1874. [6] J.J. Na¨ ppi, P.B. Dean, Mammographic feature generator for evaluation of image analysis algorithms, in: K.M. Hanson (Ed.), SPIE Medical Imaging,1997:, Image Processing, Newport Beach, USA, 1997, pp. 911 –918. [7] M.J. Lado, P.B. Tahoces, M. Souto, A.J. Mendez, J.J. Vidol, Real and simulated clustered microcalcifications in digital mammograms: ROC study of observer performance, Medical Physics 24 (1997) 1385 – 1394. [8] M. Kallergi, M.A. Gavrielides, L. He, C.G. Bernam, J.J. Kim, R.A. Clark, Simulation model of mammographic calcifications based on the American College of Radiology Breast Imaging Reporting and Data System, or BIRADS, Academic Radiology 5 (1998) 670 –679. [9] J. Hsu, D.M. Chelberg, C.F. Babbs, Z. Pizlo, E.J. Delp,
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Algorithmic 3D simulation of breast calcifications for digital mammography Janne Na¨ppi a,*, Peter B. Dean b, Olli Nevalainen a, Sakari Toikkanen c a
Department of Computer Science, Uni6ersity of Turku, Turku Centre for Computer Science (TUCS), Lemmink aisenkatu 14 A FIN-20520, Turku, Finland b Department of Diagnostic Radiology, Uni6ersity of Turku, Medical Imaging Center, Uni6ersity Central Hospital, FIN-20520, Turku, Finland c Department of Pathology, Uni6ersity Central Hospital, FIN-20520, Turku, Finland
Abstract We present a framework for algorithmic three-dimensional simulation of breast calcifications. The simulated calcifications can be viewed from any angle at a higher spatial resolution than currently available for digital mammography, and they can be placed onto a simulated or real mammographic background to provide example cases for computers and radiologists. In order to simulate calcification clusters, we also show how to simulate duct networks and terminal ductal lobular units. We evaluated the model with a double-blind evaluation of 60 cases with four experienced radiologists by mixing 30 cases of simulated calcification clusters on a real or simulated mammographic background with 30 cases of real breast calcification clusters digitized at a spatial resolution of 15 mm from high-resolution radiographs of 5 mm slices of breast specimens. The results indicate that the majority of the 2D projections of the 3D simulated calcifications compare favorably with the radiographic images of real breast calcifications. © 2001 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Breast calcifications; Simulation; Computer-assisted diagnosis; Digital mammography
1. Introduction Fewer than 20% of the cases with clustered breast calcifications are produced by malignant diseases. To identify the suspicious cases and to prevent unnecessary interventional procedures, a detailed and reliable analysis of the mammographic appearance of the calcifications is required [1–3]. * Corresponding author. E-mail address: [email protected] (J. Na¨ppi).
Computer-assisted diagnosis (CAD) [4] may be used to improve the speed and accuracy of the analysis of calcifications, but the enormous number of variations in the appearance of the calcifications on a mammogram poses a serious challenge. Ideally, the design and evaluation of the computer algorithms should be based upon carefully documented cases in which the calcifications seen on histopathologic examination are the same as those demonstrated mammographically. Since arranging this kind of documentation is extremely time-consuming, most mammographic
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images of breast calcifications are fairly imperfect representations of the actual pathological entities, and the limited spatial resolution of digital images may introduce further degradation. Furthermore, the continuous improvement in the quality of digital mammograms makes current digital archives of mammographic cases soon outdated, so they cannot be considered as a fully reliable reference material for the important task of developing and evaluating CAD. Simulation models that generate synthetic images of lesions in clinically relevant detail provide an alternative source for mammographic images. A particular advantage of the simulation models is that the image formation is fully controlled and exactly reproducible. The previous computer models for simulating breast calcifications were limited to 2D [5–8], and an inherent problem of these models was that they tried to reproduce the 2D-projections of objects that are actually 3D entities. In contrast, the 3D models presented in this paper can be based directly on the knowledge of anatomical features and the pathological processes through which real breast calcifications are produced. Other work on mammographic 3D simulation has only considered the simulation of complete breast at a low spatial resolution [9– 11]. As far as we know, our work on the algorithmic 3D simulation of breast calcifications has been unique. The major parts of our model are: (1) simulation of the X-ray image acquisition (Section 2.1); (2) algorithmic 3D simulation of the calcification particles of specified type (Section 2.2); (3) 3D simulation of the breast structures associated with the calcifications (Section 2.3); and (4) placing of the calcifications onto a real or simulated mammographic background (Section 3). The evaluation of the simulation model is presented in Section 4, with discussion and conclusions in Sections 5 and 6. The simulated calcifications can be applied to the training and evaluation of CAD algorithms and radiologists, but an analysis of these applications is beyond the scope of this paper and will be considered elsewhere. The software associated with the model has been implemented using Khoros [12].
2. The simulation model
2.1. X-ray image acquisition The X-ray image acquisition model is based on the principles of the actual physical process [13]. There are three components: an X-ray field, a simulated breast 6olume, and a film plane (Fig. 1). For each pixel in the film plane, the X-rays are emitted from the X-ray source through the simulated breast volume onto the film plane. The X-ray field is represented as an image pair (E( ,P( ), where the energy field E( specifies the current X-ray energy and the position field P( specifies the current distance to the film plane. Each pixel of E( is initialized to the X-ray energy for which the maximal transferred X-ray energy at the film plane will be 1 unit, and each pixel of P( is initialized to the distance between the X-ray source and the film plane. The simulated volume may contain a calcification surrounded by homogeneous breast tissue, a breast structure (i.e. duct network or a terminal ductal lobular unit) with calcifications, or a complete simulated breast with breast parenchyma, fatty tissue, breast structures, and calcifications (Fig. 2). The value of each voxel within the volume specifies the linear attenuation coefficient (LAC), i.e. the relative radiographic density of the matter at that position. For a homogeneous object with LAC m and thickness x, the amount of X-ray attenuation can be computed using Beer’s law of photon absorption [13]:
Fig. 1. The X-ray image acquisition model. The X-ray field is represented as an energy-position image pair (E( ,P( ).
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Fig. 2. The breast model is composed of independent modules, each of which may be imaged separately.
N =N0e − vx
(1)
Here N0 is the number of the incident X-ray photons and N the number of transmitted X-ray photons. Normal biological variation is simulated by allowing a small random variation of the LAC. The digital images of the model are represented nominally at a spatial resolution of 10 mm with 12-bit pixels. Specification of an explicit nominal resolution is not really necessary, but it greatly simplifies the application of the simulation model. However, although the high spatial resolution produces realistic results, the density modeling and computation of the X-ray penetration quickly become a computational burden as we increase the size of the simulated volume. Therefore we minimize the use of volumes by approximating the X-ray penetration mathematically in the space between the simulated volumes using Eq. (1) and the geometric relations of the breast model: the X-ray energy E(m, n) is decreased according to the LAC of the material through which the X-ray currently traverses, and the position value P(m, n) is decreased according to the traversed distance. At the film plane we have P(m, n) =0 for all positions (m, n). The film response D(m, n) is modeled by ([14], p.273) D(m, n)=k log10(− (m, n)) − d0
(2)
where g is the gamma of the film, (m, n) is the incident photon intensity at (m, n), and d0 is an optical intensity constant. The effects of film noise, scattering and other image noise are modeled simply by convolving the final X-ray energy field with a 2D Gaussian
G(m, n)=
1
2y| 2
− (m2 + n2)
e
2|2 −
(3)
where s controls the magnitude of the noise. The effect of digitization is simulated by computing the local mean of the image and downsampling it to the final spatial resolution using bilinear interpolation.
2.2. Calcification models The models of calcifications and breast structures are based on their anatomic properties that have been described in the literature [1], and observed in practice using preoperative mammograms, specimen radiographs, stereomicroscopy, and histopathologic images. We have collected this knowledge in our experimental differential diagnosis system which is similar to (but more detailed than) the American College of Radiology Breast Imaging Reporting and Data System (BIRADS). We have developed three principal models for modeling calcification particles: irregular, o6oid, and elongated. The other types of calcifications can be derived from these basic types with appropriate parameter setting and application of the associated breast structures. The calcifications can be viewed from a specified direction by rotating the calcifications in 3D ([15], pp. 223–224) before projecting the X-rays through the 3D LAC volume modeling the calcification. Irregular particles are simulated using a 3D random-walk model ([16], pp. 232–255). The model is controlled by two parameters: shape asymmetry [0,1] and boundary roughness
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[0.1]. First all voxels within the target volume V are initialized to the LAC of fatty tissue. Next the calcification is constructed by starting N()[5, 20] (N() =5 + 15) random-walk processes from the center of V. Each process assigns the LAC of calcium to the value of the current voxel, and then moves randomly to one of the neighboring voxels. Each process continues until it reaches the boundary of the surrounding volume, or until it has assigned V P() (P()[0.01, 0.125]) voxels to the calcification. At this stage the particles are rather sparse cloudy objects. Indeed, such particles can be used to simulate powderish clusters. To produce standard solid particles, we apply D() [2, 10] morphological dilations to the calcification using a spherical morphological element. Thus the model simulates the apparently irregular growth of the malignant types of calcifications. It is also simple, fast, and produces realistic results. Ovoid types of calcifications usually develop within lobules. The simulation of these mostly round particles is based on modifying the shape of an ellipsoid. The simplest case is the ovoid solid type (pearl), which is implemented by filling the ellipsoid with the LAC of calcium. Shape deformation is introduced by varying the axis ratio of the ellipsoid. Eggshell type is computed similarly, except that now only the surface region of specified thickness represents the calcium while the density of the center region is set to a LAC between that of calcium and fatty tissue. Milk-of-
calcium type is derived from the eggshell model by removing the top of the eggshell with a large sphere. Some powderish ‘sediment’ of high density is added randomly to the bottom region of the teacup. Simulation of elongated types is based on a wireframe model which is similar to the duct models discussed in Section 2.3. The wireframe defines the overall shape of the calcification, with the calcification residing within a specified radius from the wireframe. Various types of branching shapes can be generated similarly. For example, the plasma cell mastitis types are mostly quite straight and smooth, while the (solid) casting types are more twisting. The intraductal case is implemented as a solid wireframe, and the periductal case is implemented as a hollow wireframe. Examples of the simulated irregular, ovoid, and elongated particles are seen in Fig. 3. Some of the simulated calcifications may look overly regular, and although such calcifications appear in real cases (in high-quality images), it is sometimes necessary to simulate calcifications that look less than perfect. Then we apply a degradation algorithm to improve the realism of the simulated calcifications. The algorithm squeezes the calcification with a plane perpendicular to the origin by moving the intersected calcification voxels to the next plane closer to the origin (Fig. 4). The process roughly simulates the internal tissue forces that limit the growth of the calcification. The parameter 1 [0, 1] characterizes the maxi-
Fig. 3. Examples of simulated irregular (top), ovoid (middle), and elongated (bottom) particles.
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Fig. 4. The first steps of the degradation algorithm (top): the particle is squeezed by a plane, rotated to a random orientation, and squeezed again. Bottom row: a perfectly round calcification (left), and results of the degradation algorithm.
mum relative distance which the squeezing plane can move from the surface towards the origin. The squeezing is repeated for o(2) [9, 20] (2 [0, 1]) random orientations. Despite its simplicity, the algorithm produces visually good results.
2.3. Models for breast structures and mammographic background The growth pattern of most calcification clusters relates to the location and shape of the breast structures within which the calcification particles appear. The most important breast structures in this respect are the duct network and the terminal ductal lobular units (TDLUs). Modelling of these structures is necessary for getting a detailed simulation of 3D calcification clusters. Visualization of biological ducts ([17,18], p. 30; 3, pp. A4 – A6) shows tree-like patterns which can be represented compactly by fractal geometry [10,11,19]. In most cases we use a random midpoint displacement algorithm [16]. The algorithm is initially applied to a two-point (poly)line P0 = [(x00, y00, z00), (x01, y01, z01)] which roughly defines the ultimate length and orientation of the final duct network. The algorithm then splits P0 at midpoint to produce a new polyline P1 =F(P0) = [(x10, y10, z10), (x11, y11, z11), (x12, y12, z12)]. By repeating this procedure to the prespecified precision we obtain a polyline L0 =Pn =F(Pn -1), which is then expanded to the next level by allowing some of the points (xni, yni, zni) to branch in up to three new ducts (i.e. polylines). The al-
gorithm is applied recursively to each of the new ducts, until the duct system has been expanded to the specified number of levels (Fig. 5). The final duct network consists of a set L= (Li ) of connected polylines. The general form of L is controlled by the limiting bounds of the branching angle and the length of the polyline segments. The generation of the calcifications within the duct is controlled by associating attributes with the nodes of the polyline. A particularly useful attribute is the duct diameter, which for example allows simulation of the cases where cancer seems to spread from the ‘buds’ towards the main duct in the form of malignant calcifications. Ramification matrices are a useful alternative for generating the duct network [11,20]. In short, the matrix specifies the branching probabilities that generate the topology of a binary tree system. We can control the overall shape of the resulting duct system by introducing length, diameter, and deviation attributes which depend on the order of the nodes of the tree [20]. The flexible control over the final form makes ramification matrices especially useful for simulating the duct network of a complete breast, although several matrices and attribute functions are needed for truly realistic results. It is not necessary to place the simulated duct network into an actual volume, since we can place the calcifications simply by tracing the polyline representation and constructing the appropriate wireframe models for generating the individual calcification particles.
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Fig. 5. Examples of simulated duct networks. Top left: demonstration of the midpoint replacement algorithm. Bottom left: examples of the resulting 3D duct networks. Right: a duct network generated by two ramification matrices of a random binary tree [11, 20].
A TDLU is a terminal part of the duct system which consists of a large number of lobules, making it resemble a bunch of grapes. The calcifications develop within the lobules, and we only need to know the lobule coordinates in order to place the calcifications. To generate a TDLU, we model the lobules as spheres which are placed around the origin in a series of layers (Fig. 6), in which the distance between the neighboring spheres is minimized. The complete breast simulation models reported so far, including our own, cannot yet produce a truly realistic variable mammographic background. However, in most cases the simulation of calcifications only requires relatively small background regions, so we are currently using two simplified sources for mammographic background: real mammographic images and fractal texture. The fractal texture generator that we use produces visually realistic mammographic background. It is based on a recursi6e 2D midpoint displacement algorithm [16], in which at each step the current image is interpreted as a square lattice of pixels. After adding a new pixel to the midpoint of every lattice square, the image is rotated by 45 degrees to obtain a new image with a higher spatial resolution. The value of each new pixel is computed using the average of the surrounding pixels, the specified standard deviation, and the specified Hurst coefficient H [0,1]. Visual com-
parison between the fractal texture and real mammographic texture suggests that the texture generated with Hurst coefficients 0.1B HB 0.5 is most realistic (Fig. 7). For H\ 0.5 the image ‘noise’ becomes overly minimal. The fractal texture generator can only produce background that seemingly combines fatty tissue and breast parenchyma. Other distinctive breast features, such as ducts, Cooper’s ligaments, or tumors, should be modeled separately.
3. Placing the calcifications onto mammographic background To place the calcifications, it is usually convenient to use a contrast parameter Ci for character-
Fig. 6. A large TDLU being generated (left), and a small TDLU in which the lobule positions have been randomly varied. The spheres represent the maximal space available for each lobule.
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Fig. 7. Examples of simulated mammographic background. From left to right: H =0.20, H =0.33, H = 0.50, and H =0.75.
izing the level of embedding for each calcification i. For realistic results the contrast should vary slightly between each calcification of a cluster. A basic method is to compute the mean of the calcification (I mean ) and its target background c (I mean bg ),and then scale the intensities of the calcification to satisfy (I mean ) = I mean +I mean c bg bg Ci. Here we could also use maximum intensities and/or standard deviations. Other techniques include adjustment of the intensities in concentric rings based on a statistical analysis of a large number of real cases [5], and scaling of the image intensities according to the relative intensities of calcium and fatty tissue deduced from the background. All these techniques are applicable but produce somewhat different results. The 3D simulated calcifications can be placed onto mammographic background using these methods. However, the methods were originally developed for use with preoperative mammograms, and in specimen radiographs the results are less satisfactory because of the improved image quality and the large size variation of the calcifications. For example, some of the methods only modify the boundary region of the calcification, and the inconsistency between the center region and the background may be clearly visible. Here we use a novel method for combining 3D simulated calcifications with mammographic background. The idea is to interpret the mammographic background as a representation of the relative densities of the original mammographic
volume. First we simulate the X-ray penetration through the 3D calcifications, and then add the attenuation caused by the LACs derived from the background. The contrast of the calcification is controlled by varying the relative magnitude of the attenuation. Since the calcifications may nevertheless look too smooth and regular compared with the background, we can add random variation to the X-ray field coefficients relative to the standard deviation of the background intensities. Examples of placing a calcification onto mammographic background are seen in Fig. 8.
Fig. 8. Top: ideal calcification (left) is degraded (middle), and mammographic noise is added (right). Middle: simulated background before (left) and after placing the calcification (middle and right). Bottom: example results of our new method.
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Table 1 Results of four radiologists’ evaluation of 30 simulated and 30 real breast calcifications Summary of the radiologists’ evaluation of the calcifications
Correctly evaluated
Incorrectly evaluated
Undecided
Real calcifications (n =30) Simulated calcifications (n= 30) All calcifications (n = 60)
62% 54% 58%
19% 22% 20%
19% 24% 22%
4. Results We evaluated the simulation model by digitizing 50 specimen radiographs of thin (5 mm) breast specimens at a spatial resolution of 15 mm with a 10-bit intensity resolution, and extracting thirty 512*512-regions of interest containing calcifications. The extracted regions could contain both benign and malignant calcifications. Next we generated 30 cases of 3D simulated calcification clusters, and placed them onto both real (15 cases) and simulated (15 cases) mammographic background. The cases of real mammographic background did not contain any real calcifications. The original 10-bit images were scaled to 8-bit images to present them on a standard 17’’ PC computer screen. We performed the evaluation by showing randomly mixed real and simulated regions of interest with calcifications, one at a time, to four radiologists with considerable (10– 25 years) experience in interpreting mammograms. A questionnaire containing the following statement was presented to each radiologist: Some of these images contain real calcification(s) from specimen radiographs. The other images contain computer simulated calcification(s). Please evaluate each image according to whether the calcifications appear real or simulated. 0) No calcifications 1) Definitely real calcification(s) 2) Probably real calcification(s) 3) Could be either 4) Probably simulated calcification(s) 5) Definitely simulated calcification(s) The four radiologists differed in their evaluation according to their experience with the speci-
men radiographs on which the evaluation material was based, those with more experience having higher accuracy. As can be seen from Table 1, the real calcifications were only slightly more accurately evaluated than were the simulated calcifications. There was a greater interobserver variation with the simulated calcifications. The presence of a real or simulated mammographic tissue background had no effect upon the evaluation. Eight of the cases of simulated calcifications were correctly considered to be simulated by all of the radiologists, and in four of the cases of simulated calcifications none of the radiologists recognized them as simulated. A majority (18/30) of the simulated calcifications was considered to be real by at least one of the radiologists (Examples, Fig. 9).
5. Discussion The results of the radiologists’ evaluation indicate that some, but not all, of the examples of simulated calcification clusters do indeed appear as real calcifications. The results will help us to select those parameters that produce calcifications most reliably resembling true breast calcifications. For example, the preliminary analysis suggests that the realism of the simulated calcification clusters could be generally improved by increasing the size, shape and density variance of the particles, and by generating larger clusters. The evaluation was particularly demanding as the simulated calcifications were compared with very high quality, high-resolution (15 mm) digitized images of thin (5 mm) breast slices.
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6. Conclusion Algorithmically 3D simulated calcifications are an alternative to real mammographic data to produce cases for evaluating image processing algorithms and for teaching computers and radiologists. Images generated by 3D models are conceptually superior to those generated by 2D models: anatomic knowledge is readily applicable, the image formation is fully controlled, and the simulated calcifications can be placed onto any mammographic background. The results suggest that the more successful 2D projections of the 3D simulated calcifications compare favorably with high resolution (15 mm) specimen radiographic images of real breast cal-
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cifications. The more successful parameters are undergoing further development. Future work on the algorithmically 3D simulated calcifications involves development of a simpler user interface and further adjustment of the models, as well as their application in computerassisted diagnosis. A new evaluation study with a larger number of cases and a larger amount of highly realistic simulated cases than that used here is also planned.
Acknowledgements This work was supported by grants from the Instrumentarium Science Foundation, EVO Spe-
Fig. 9. Four distinct examples of the mammographic clusters of the evaluation image set. Top row: two real cases, labeled as real (left) and simulated (right) by all radiologists. Bottom row: two simulated cases, labeled as real (left) and simulated (right) by all radiologists.
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cial Funds, Turku University Foundation, and by the Finnish Academy. We would also like to thank the radiologists who participated in the evaluation.
[10]
[11]
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