Power-law and log-normal avalanche size statistics in random growth processes.

We study the avalanche statistics observed in a minimal random growth model. The growth is governed by a reproduction rate obeying a probability distribution with finite mean a[over ¯] and variance v_{a}. These two control parameters determine if the avalanche size tends to a stationary distribution (finite scale statistics with finite mean and variance, or power-law tailed statistics with exponent ∈(1,3]), or instead to a nonstationary regime with log-normal statistics. Numerical results and their statistical analysis are presented for a uniformly distributed growth rate, which are corroborated and generalized by mathematical results. The latter show that the numerically observed avalanche regimes exist for a wide family of growth rate distributions, and they provide a precise definition of the boundaries between the three regimes.

Loss of Mammographic Tissue Homeostasis in Invasive Lobular and Ductal Breast Carcinomas vs. Benign Lesions.

The 2D wavelet transform modulus maxima (WTMM) method is used to perform a comparison of the spatial fluctuations of mammographic breast tissue from patients with invasive lobular carcinoma, those with invasive ductal carcinoma, and those with benign lesions. We follow a procedure developed and validated in a previous study, in which a sliding window protocol is used to analyze thousands of small subregions in a given mammogram. These subregions are categorized according to their Hurst exponent values (H): fatty tissue (H ≤ 0.45), dense tissue (H ≥ 0.55), and disrupted tissue potentially linked with tumor-associated loss of homeostasis (0.45 < H < 0.55). Following this categorization scheme, we compare the mammographic tissue composition of the breasts. First, we show that cancerous breasts are significantly different than breasts with a benign lesion (p-value ∼ 0.002). Second, the asymmetry between a patient's cancerous breast and its contralateral counterpart, when compared to the asymmetry from patients with benign lesions, is also statistically significant (p-value ∼ 0.006). And finally, we show that lobular and ductal cancerous breasts show similar levels of disruption and similar levels of asymmetry. This study demonstrates reproducibility of the WTMM sliding-window approach to help detect and characterize tumor-associated breast tissue disruption from standard mammography. It also shows promise to help with the detection lobular lesions that typically go undetected via standard screening mammography at a much higher rate than ductal lesions. Here both types are assessed similarly.

Multifractal Desynchronization of the Cardiac Excitable Cell Network During Atrial Fibrillation. II. Modeling.

In a companion paper (I. Multifractal analysis of clinical data), we used a wavelet-based multiscale analysis to reveal and quantify the multifractal intermittent nature of the cardiac impulse energy in the low frequency range ≲ 2Hz during atrial fibrillation (AF). It demarcated two distinct areas within the coronary sinus (CS) with regionally stable multifractal spectra likely corresponding to different anatomical substrates. The electrical activity also showed no sign of the kind of temporal correlations typical of cascading processes across scales, thereby indicating that the multifractal scaling is carried by variations in the large amplitude oscillations of the recorded bipolar electric potential. In the present study, to account for these observations, we explore the role of the kinetics of gap junction channels (GJCs), in dynamically creating a new kind of imbalance between depolarizing and repolarizing currents. We propose a one-dimensional (1D) spatial model of a denervated myocardium, where the coupling of cardiac cells fails to synchronize the network of cardiac cells because of abnormal transjunctional capacitive charging of GJCs. We show that this non-ohmic nonlinear conduction 1D modeling accounts quantitatively well for the "multifractal random noise" dynamics of the electrical activity experimentally recorded in the left atrial posterior wall area. We further demonstrate that the multifractal properties of the numerical impulse energy are robust to changes in the model parameters.

Developmental and cancer-associated plasticity of DNA replication preferentially targets GC-poor, lowly expressed and late-replicating regions.

The spatiotemporal program of metazoan DNA replication is regulated during development and altered in cancers. We have generated novel OK-seq, Repli-seq and RNA-seq data to compare the DNA replication and gene expression programs of twelve cancer and non-cancer human cell types. Changes in replication fork directionality (RFD) determined by OK-seq are widespread but more frequent within GC-poor isochores and largely disconnected from transcription changes. Cancer cell RFD profiles cluster with non-cancer cells of similar developmental origin but not with different cancer types. Importantly, recurrent RFD changes are detected in specific tumour progression pathways. Using a model for establishment and early progression of chronic myeloid leukemia (CML), we identify 1027 replication initiation zones (IZs) that progressively change efficiency during long-term expression of the BCR-ABL1 oncogene, being twice more often downregulated than upregulated. Prolonged expression of BCR-ABL1 results in targeting of new IZs and accentuation of previous efficiency changes. Targeted IZs are predominantly located in GC-poor, late replicating gene deserts and frequently silenced in late CML. Prolonged expression of BCR-ABL1 results in massive deletion of GC-poor, late replicating DNA sequences enriched in origin silencing events. We conclude that BCR-ABL1 expression progressively affects replication and stability of GC-poor, late-replicating regions during CML progression.

The eukaryotic bell-shaped temporal rate of DNA replication origin firing emanates from a balance between origin activation and passivation.

The time-dependent rate [Formula: see text] of origin firing per length of unreplicated DNA presents a universal bell shape in eukaryotes that has been interpreted as the result of a complex time-evolving interaction between origins and limiting firing factors. Here, we show that a normal diffusion of replication fork components towards localized potential replication origins (p-oris) can more simply account for the [Formula: see text] universal bell shape, as a consequence of a competition between the origin firing time and the time needed to replicate DNA separating two neighboring p-oris. We predict the [Formula: see text] maximal value to be the product of the replication fork speed with the squared p-ori density. We show that this relation is robustly observed in simulations and in experimental data for several eukaryotes. Our work underlines that fork-component recycling and potential origins localization are sufficient spatial ingredients to explain the universality of DNA replication kinetics.

Prestressed cells are prone to cytoskeleton failures under localized shear strain: an experimental demonstration on muscle precursor cells.

We report on a wavelet based space-scale decomposition method for analyzing the response of living muscle precursor cells (C2C12 myoblasts and myotubes) upon sharp indentation with an AFM cantilever and quantifying their aptitude to sustain such a local shear strain. Beyond global mechanical parameters which are currently used as markers of cell contractility, we emphasize the necessity of characterizing more closely the local fluctuations of the shear relaxation modulus as they carry important clues about the mechanisms of cytoskeleton strain release. Rupture events encountered during fixed velocity shear strain are interpreted as local disruptions of the actin cytoskeleton structures, the strongest (brittle) ones being produced by the tighter and stiffer stress fibers or actin agglomerates. These local strain induced failures are important characteristics of the resilience of these cells, and their aptitude to maintain their shape via a quick recovery from local strains. This study focuses on the perinuclear region because it can be considered as a master mechanical organizing center of these muscle precursor cells. Using this wavelet-based method, we combine the global and local approaches for a comparative analysis of the mechanical parameters of normal myoblasts, myotubes and myoblasts treated with actomyosin cytoskeleton disruptive agents (ATP depletion, blebbistatin).

Evidence for DNA Sequence Encoding of an Accessible Nucleosomal Array across Vertebrates.

Nucleosome-depleted regions around which nucleosomes order following the "statistical" positioning scenario were recently shown to be encoded in the DNA sequence in human. This intrinsic nucleosomal ordering strongly correlates with oscillations in the local GC content as well as with the interspecies and intraspecies mutation profiles, revealing the existence of both positive and negative selection. In this letter, we show that these predicted nucleosome inhibitory energy barriers (NIEBs) with compacted neighboring nucleosomes are indeed ubiquitous to all vertebrates tested. These 1 kb-sized chromatin patterns are widely distributed along vertebrate chromosomes, overall covering more than a third of the genome. We have previously observed in human deviations from neutral evolution at these genome-wide distributed regions, which we interpreted as a possible indication of the selection of an open, accessible, and dynamic nucleosomal array to constitutively facilitate the epigenetic regulation of nuclear functions in a cell-type-specific manner. As a first, very appealing observation supporting this hypothesis, we report evidence of a strong association between NIEB borders and the poly(A) tails of Alu sequences in human. These results suggest that NIEBs provide adequate chromatin patterns favorable to the integration of Alu retrotransposons and, more generally to various transposable elements in the genomes of primates and other vertebrates.

Multi-scale structural community organisation of the human genome.

BACKGROUND: Structural interaction frequency matrices between all genome loci are now experimentally achievable thanks to high-throughput chromosome conformation capture technologies. This ensues a new methodological challenge for computational biology which consists in objectively extracting from these data the structural motifs characteristic of genome organisation. RESULTS: We deployed the fast multi-scale community mining algorithm based on spectral graph wavelets to characterise the networks of intra-chromosomal interactions in human cell lines. We observed that there exist structural domains of all sizes up to chromosome length and demonstrated that the set of structural communities forms a hierarchy of chromosome segments. Hence, at all scales, chromosome folding predominantly involves interactions between neighbouring sites rather than the formation of links between distant loci. CONCLUSIONS: Multi-scale structural decomposition of human chromosomes provides an original framework to question structural organisation and its relationship to functional regulation across the scales. By construction the proposed methodology is independent of the precise assembly of the reference genome and is thus directly applicable to genomes whose assembly is not fully determined.

Mammographic evidence of microenvironment changes in tumorous breasts.

PURPOSE: The microenvironment of breast tumors plays a critical role in tumorigenesis. As long as the structural integrity of the microenvironment is upheld, the tumor is suppressed. If tissue structure is lost through disruptions in the normal cell cycle, the microenvironment may act as a tumor promoter. Therefore, the properties that distinguish between healthy and tumorous tissues may not be solely in the tumor characteristics but rather in surrounding non-tumor tissue. The goal of this paper was to show preliminary evidence that tissue disruption and loss of homeostasis in breast tissue microenvironment and breast bilateral asymmetry can be quantitatively and objectively assessed from mammography via a localized, wavelet-based analysis of the whole breast. METHODS: A wavelet-based multifractal formalism called the 2D Wavelet Transform Modulus Maxima (WTMM) method was used to quantitate density fluctuations from mammographic breast tissue via the Hurst exponent (H). Each entire mammogram was cut in hundreds of 360 × 360 pixel subregions in a gridding scheme of overlapping sliding windows, with each window boundary separated by 32 pixels. The 2D WTMM method was applied to each subregion individually. A data mining approach was set up to determine which metrics best discriminated between normal vs. cancer cases. These same metrics were then used, without modification, to discriminate between normal vs. benign and benign vs. cancer cases. RESULTS: The density fluctuations in healthy mammographic breast tissue are either monofractal anti-correlated (H < 1/2) for fatty tissue or monofractal long-range correlated (H>1/2) for dense tissue. However, tissue regions with H~1/2, as well as left vs. right breast asymetries, were found preferably in tumorous (benign or cancer) breasts vs. normal breasts, as quantified via a combination metric yielding a P-value ~ 0.0006. No metric considered showed significant differences between cancer vs. benign breasts. CONCLUSIONS: Since mammographic tissue regions associated with uncorrelated (H~1/2) density fluctuations were predominantly in tumorous breasts, and since the underlying physical processes associated with a H~1/2 signature are those of randomness, lack of spatial correlation, and free diffusion, it is hypothesized that this signature is also associated with tissue disruption and loss of tissue homeostasis.

Multifractal Desynchronization of the Cardiac Excitable Cell Network During Atrial Fibrillation. I. Multifractal Analysis of Clinical Data.

Atrial fibrillation (AF) is a cardiac arrhythmia characterized by rapid and irregular atrial electrical activity with a high clinical impact on stroke incidence. Best available therapeutic strategies combine pharmacological and surgical means. But when successful, they do not always prevent long-term relapses. Initial success becomes all the more tricky to achieve as the arrhythmia maintains itself and the pathology evolves into sustained or chronic AF. This raises the open crucial issue of deciphering the mechanisms that govern the onset of AF as well as its perpetuation. In this study, we develop a wavelet-based multi-scale strategy to analyze the electrical activity of human hearts recorded by catheter electrodes, positioned in the coronary sinus (CS), during episodes of AF. We compute the so-called multifractal spectra using two variants of the wavelet transform modulus maxima method, the moment (partition function) method and the magnitude cumulant method. Application of these methods to long time series recorded in a patient with chronic AF provides quantitative evidence of the multifractal intermittent nature of the electric energy of passing cardiac impulses at low frequencies, i.e., for times (≳0.5 s) longer than the mean interbeat (≃ 10-1 s). We also report the results of a two-point magnitude correlation analysis which infers the absence of a multiplicative time-scale structure underlying multifractal scaling. The electric energy dynamics looks like a "multifractal white noise" with quadratic (log-normal) multifractal spectra. These observations challenge concepts of functional reentrant circuits in mechanistic theories of AF, still leaving open the role of the autonomic nervous system (ANS). A transition is indeed observed in the computed multifractal spectra which group according to two distinct areas, consistently with the anatomical substrate binding to the CS, namely the left atrial posterior wall, and the ligament of Marshall which is innervated by the ANS. In a companion paper (II. Modeling), we propose a mathematical model of a denervated heart where the kinetics of gap junction conductance alone induces a desynchronization of the myocardial excitable cells, accounting for the multifractal spectra found experimentally in the left atrial posterior wall area.

Comparative Multifractal Analysis of Dynamic Infrared Thermograms and X-Ray Mammograms Enlightens Changes in the Environment of Malignant Tumors.

There is growing evidence that the microenvironment surrounding a tumor plays a special role in cancer development and cancer therapeutic resistance. Tumors arise from the dysregulation and alteration of both the malignant cells and their environment. By providing tumor-repressing signals, the microenvironment can impose and sustain normal tissue architecture. Once tissue homeostasis is lost, the altered microenvironment can create a niche favoring the tumorigenic transformation process. A major challenge in early breast cancer diagnosis is thus to show that these physiological and architectural alterations can be detected with currently used screening techniques. In a recent study, we used a 1D wavelet-based multi-scale method to analyze breast skin temperature temporal fluctuations collected with an IR thermography camera in patients with breast cancer. This study reveals that the multifractal complexity of temperature fluctuations superimposed on cardiogenic and vasomotor perfusion oscillations observed in healthy breasts is lost in malignant tumor foci in cancerous breasts. Here we use a 2D wavelet-based multifractal method to analyze the spatial fluctuations of breast density in the X-ray mammograms of the same panel of patients. As compared to the long-range correlations and anti-correlations in roughness fluctuations, respectively observed in dense and fatty breast areas, some significant change in the nature of breast density fluctuations with some clear loss of correlations is detected in the neighborhood of malignant tumors. This attests to some architectural disorganization that may deeply affect heat transfer and related thermomechanics in breast tissues, corroborating the change to homogeneous monofractal temperature fluctuations recorded in cancerous breasts with the IR camera. These results open new perspectives in computer-aided methods to assist in early breast cancer diagnosis.

Evidence of selection for an accessible nucleosomal array in human.

BACKGROUND: Recently, a physical model of nucleosome formation based on sequence-dependent bending properties of the DNA double-helix has been used to reveal some enrichment of nucleosome-inhibiting energy barriers (NIEBs) nearby ubiquitous human "master" replication origins. Here we use this model to predict the existence of about 1.6 millions NIEBs over the 22 human autosomes. RESULTS: We show that these high energy barriers of mean size 153 bp correspond to nucleosome-depleted regions (NDRs) in vitro, as expected, but also in vivo. On either side of these NIEBs, we observe, in vivo and in vitro, a similar compacted nucleosome ordering, suggesting an absence of chromatin remodeling. This nucleosomal ordering strongly correlates with oscillations of the GC content as well as with the interspecies and intraspecies mutation profiles along these regions. Comparison of these divergence rates reveals the existence of both positive and negative selections linked to nucleosome positioning around these intrinsic NDRs. Overall, these NIEBs and neighboring nucleosomes cover 37.5 % of the human genome where nucleosome occupancy is stably encoded in the DNA sequence. These 1 kb-sized regions of intrinsic nucleosome positioning are equally found in GC-rich and GC-poor isochores, in early and late replicating regions, in intergenic and genic regions but not at gene promoters. CONCLUSION: The source of selection pressure on the NIEBs has yet to be resolved in future work. One possible scenario is that these widely distributed chromatin patterns have been selected in human to impair the condensation of the nucleosomal array into the 30 nm chromatin fiber, so as to facilitate the epigenetic regulation of nuclear functions in a cell-type-specific manner.

Time-lapse scanning surface plasmon microscopy of living adherent cells with a radially polarized beam.

We report on a fibered high-resolution scanning surface plasmon microscope for long term imaging of living adherent cells. The coupling of a high numerical aperture objective lens and a fibered heterodyne interferometer enhances both the sensitivity and the long term stability of this microscope, allowing for time-lapse recording over several days. The diffraction limit is reached with a radially polarized illumination beam. Adherence and motility of living C2C12 myoblast cells are followed for 50 h, revealing that the dynamics of these cells change after 10 h. This plasmon enhanced evanescent wave microscopy is particularly suited for investigating cell adhesion, since it can not only be performed without staining of the sample but it can also capture in real time the exchange of extracellular matrix elements between the substrate and the cells.

Deciphering the internal complexity of living cells with quantitative phase microscopy: a multiscale approach.

The distribution of refractive indices (RIs) of a living cell contributes in a nonintuitive manner to its optical phase image and quite rarely can be inverted to recover its internal structure. The interpretation of the quantitative phase images of living cells remains a difficult task because (1) we still have very little knowledge on the impact of its internal macromolecular complexes on the local RI and (2) phase changes produced by light propagation through the sample are mixed with diffraction effects by the internal cell bodies. We propose to implement a two-dimensional wavelet-based contour chain detection method to distinguish internal boundaries based on their greatest optical path difference gradients. These contour chains correspond to the highest image phase contrast and follow the local RI inhomogeneities linked to the intracellular structural intricacy. Their statistics and spatial distribution are the morphological indicators suited for comparing cells of different origins and/or to follow their transformation in pathologic situations. We use this method to compare nonadherent blood cells from primary and laboratory culture origins and to assess the internal transformation of hematopoietic stem cells by the transduction of the BCR-ABL oncogene responsible for the chronic myelogenous leukemia.

Single Cell Wall Nonlinear Mechanics Revealed by a Multiscale Analysis of AFM Force-Indentation Curves.

Individual plant cells are rather complex mechanical objects. Despite the fact that their wall mechanical strength may be weakened by comparison with their original tissue template, they nevertheless retain some generic properties of the mother tissue, namely the viscoelasticity and the shape of their walls, which are driven by their internal hydrostatic turgor pressure. This viscoelastic behavior, which affects the power-law response of these cells when indented by an atomic force cantilever with a pyramidal tip, is also very sensitive to the culture media. To our knowledge, we develop here an original analyzing method, based on a multiscale decomposition of force-indentation curves, that reveals and quantifies for the first time the nonlinearity of the mechanical response of living single plant cells upon mechanical deformation. Further comparing the nonlinear strain responses of these isolated cells in three different media, we reveal an alteration of their linear bending elastic regime in both hyper- and hypotonic conditions.

Structural organization of human replication timing domains.

Recent analysis of genome-wide epigenetic modification data, mean replication timing (MRT) profiles and chromosome conformation data in mammals have provided increasing evidence that flexibility in replication origin usage is regulated locally by the epigenetic landscape and over larger genomic distances by the 3D chromatin architecture. Here, we review the recent results establishing some link between replication domains and chromatin structural domains in pluripotent and various differentiated cell types in human. We reconcile the originally proposed dichotomic picture of early and late constant timing regions that replicate by multiple rather synchronous origins in separated nuclear compartments of open and closed chromatins, with the U-shaped MRT domains bordered by "master" replication origins specified by a localized (∼200-300 kb) zone of open and transcriptionally active chromatin from which a replication wave likely initiates and propagates toward the domain center via a cascade of origin firing. We discuss the relationships between these MRT domains, topologically associated domains and lamina-associated domains. This review sheds a new light on the epigenetically regulated global chromatin reorganization that underlies the loss of pluripotency and the determination of differentiation properties.

Embryonic stem cell specific "master" replication origins at the heart of the loss of pluripotency.

Epigenetic regulation of the replication program during mammalian cell differentiation remains poorly understood. We performed an integrative analysis of eleven genome-wide epigenetic profiles at 100 kb resolution of Mean Replication Timing (MRT) data in six human cell lines. Compared to the organization in four chromatin states shared by the five somatic cell lines, embryonic stem cell (ESC) line H1 displays (i) a gene-poor but highly dynamic chromatin state (EC4) associated to histone variant H2AZ rather than a HP1-associated heterochromatin state (C4) and (ii) a mid-S accessible chromatin state with bivalent gene marks instead of a polycomb-repressed heterochromatin state. Plastic MRT regions (≲ 20% of the genome) are predominantly localized at the borders of U-shaped timing domains. Whereas somatic-specific U-domain borders are gene-dense GC-rich regions, 31.6% of H1-specific U-domain borders are early EC4 regions enriched in pluripotency transcription factors NANOG and OCT4 despite being GC poor and gene deserts. Silencing of these ESC-specific "master" replication initiation zones during differentiation corresponds to a loss of H2AZ and an enrichment in H3K9me3 mark characteristic of late replicating C4 heterochromatin. These results shed a new light on the epigenetically regulated global chromatin reorganization that underlies the loss of pluripotency and lineage commitment.

Ubiquitous human 'master' origins of replication are encoded in the DNA sequence via a local enrichment in nucleosome excluding energy barriers.

As the elementary building block of eukaryotic chromatin, the nucleosome is at the heart of the compromise between the necessity of compacting DNA in the cell nucleus and the required accessibility to regulatory proteins. The recent availability of genome-wide experimental maps of nucleosome positions for many different organisms and cell types has provided an unprecedented opportunity to elucidate to what extent the DNA sequence conditions the primary structure of chromatin and in turn participates in the chromatin-mediated regulation of nuclear functions, such as gene expression and DNA replication. In this study, we use in vivo and in vitro genome-wide nucleosome occupancy data together with the set of nucleosome-free regions (NFRs) predicted by a physical model of nucleosome formation based on sequence-dependent bending properties of the DNA double-helix, to investigate the role of intrinsic nucleosome occupancy in the regulation of the replication spatio-temporal programme in human. We focus our analysis on the so-called replication U/N-domains that were shown to cover about half of the human genome in the germline (skew-N domains) as well as in embryonic stem cells, somatic and HeLa cells (mean replication timing U-domains). The 'master' origins of replication (MaOris) that border these megabase-sized U/N-domains were found to be specified by a few hundred kb wide regions that are hyper-sensitive to DNase I cleavage, hypomethylated, and enriched in epigenetic marks involved in transcription regulation, the hallmarks of localized open chromatin structures. Here we show that replication U/N-domain borders that are conserved in all considered cell lines have an environment highly enriched in nucleosome-excluding-energy barriers, suggesting that these ubiquitous MaOris have been selected during evolution. In contrast, MaOris that are cell-type-specific are mainly regulated epigenetically and are no longer favoured by a local abundance of intrinsic NFRs encoded in the DNA sequence. At the smaller few hundred bp scale of gene promoters, CpG-rich promoters of housekeeping genes found nearby ubiquitous MaOris as well as CpG-poor promoters of tissue-specific genes found nearby cell-type-specific MaOris, both correspond to in vivo NFRs that are not coded as nucleosome-excluding-energy barriers. Whereas the former promoters are likely to correspond to high occupancy transcription factor binding regions, the latter are an illustration that gene regulation in human is typically cell-type-specific.

Wavelet-based 3D reconstruction of microcalcification clusters from two mammographic views: new evidence that fractal tumors are malignant and Euclidean tumors are benign.

The 2D Wavelet-Transform Modulus Maxima (WTMM) method was used to detect microcalcifications (MC) in human breast tissue seen in mammograms and to characterize the fractal geometry of benign and malignant MC clusters. This was done in the context of a preliminary analysis of a small dataset, via a novel way to partition the wavelet-transform space-scale skeleton. For the first time, the estimated 3D fractal structure of a breast lesion was inferred by pairing the information from two separate 2D projected mammographic views of the same breast, i.e. the cranial-caudal (CC) and mediolateral-oblique (MLO) views. As a novelty, we define the "CC-MLO fractal dimension plot", where a "fractal zone" and "Euclidean zones" (non-fractal) are defined. 118 images (59 cases, 25 malignant and 34 benign) obtained from a digital databank of mammograms with known radiologist diagnostics were analyzed to determine which cases would be plotted in the fractal zone and which cases would fall in the Euclidean zones. 92% of malignant breast lesions studied (23 out of 25 cases) were in the fractal zone while 88% of the benign lesions were in the Euclidean zones (30 out of 34 cases). Furthermore, a Bayesian statistical analysis shows that, with 95% credibility, the probability that fractal breast lesions are malignant is between 74% and 98%. Alternatively, with 95% credibility, the probability that Euclidean breast lesions are benign is between 76% and 96%. These results support the notion that the fractal structure of malignant tumors is more likely to be associated with an invasive behavior into the surrounding tissue compared to the less invasive, Euclidean structure of benign tumors. Finally, based on indirect 3D reconstructions from the 2D views, we conjecture that all breast tumors considered in this study, benign and malignant, fractal or Euclidean, restrict their growth to 2-dimensional manifolds within the breast tissue.