2D Wavelet Transform Modulus Maxima Research Articles

Abstract PO3-07-01: Multiscale Computational Analysis of Patient-Matched Multi-Modal Imaging of Breast Tissue Microenvironment

Abstract Introduction: Mammographic breast density is a strong quantitative cancer risk factor, but its link to underlying tissue and cellular level changes are weakly understood. Understanding the complex restructuring changes in the breast tumor microenvironment and its impact on breast density is integral to improving risk assessment and treatment strategies. A novel approach was developed to stratify mammographic dense breast tissue into areas of active dense tissue and passive dense tissue. Active dense tissue is structurally reorganizing and links to cancer dynamics, whereas passive dense mammographic tissue remains organized.1,2 A complementary wavelet-based analysis technique to measure the multiscale anisotropy of collagen fibers from Second-Harmonic Generation (SHG) imaging demonstrated clinical cancer diagnostic potential in mouse carcinoma and human pancreatic cancer.3,4 This tool is now leveraged for a multi-modal study in which patient-matched H&E tissue biopsies are imaged using both bright-field microscopy and SHG imaging to compare with the patient’s corresponding mammograms. Stratifying the mammographic density into active and passive dense areas, a correlation between biopsy tissue anisotropy and mammographic density subtypes can be calculated to help understand how tumor microenvironment changes affect a patient’s mammographic breast tissue composition. Methods: SHG and brightfield-imaged biopsy slides from 10 patients (five malignant, five benign) collected from Maine Medical Center (Maine, US) were analyzed using the 2D Wavelet Transform Modulus Maxima (WTMM) Anisotropy Method generating a large-scale and a small-scale anisotropy factor as well as a scale-combined anisotropy calculated from their large- and small-scale anisotropy separation. A multi-modal anisotropy score was calculated by adding the imaged area’s H&E combined anisotropy score with its respective SHG combined anisotropy score. This analysis was done on eight randomly selected ductal areas for each imaging modality on a slide and correlated with the measured areas of density subtypes from the whole mammographic view. Linear regression was used for determining correlation between H&E and SHG anisotropy factors as well as with active and passive mammographic density subtype measurements. Results: The malignant patients’ small-scale H&E anisotropy factor positively correlated with their small scale SHG anisotropy factor (R2=0.571). No correlation was found for the benign patients (R2=0.011). The strongest correlations were found between benign patients’ mammographic passive (positive correlation) and active (negative correlation) dense tissues vs the H&E anisotropy scale-combined factor (R2=0.858 and R2=0.961, respectively). The largest malignant patient correlation was a negative relationship between SHG anisotropy scale-combined factor and overall mammographic density (inclusive of both active and passive dense tissue) (R2=0.891). The multi-modal anisotropy score correlated with overall mammographic density for both benign (positive correlation, R2=0.698) and malignant patients (negative correlation, R2=0.773). Combining the multi-modal anisotropy factor from SHG and H&E with the active dense tissue measurement using K-Means clustering, 5/5 malignant and 4/5 benign patients were correctly classified. Conclusion: Density in mammograms correlates with underlying quantitative changes in a patient’s tissue and collagen anisotropy. Combining both H&E and SHG analysis on tissue biopsy with active and passive density metrics from computational mammography leads to the best predictive potential than any one method on their own.

Multiscale anisotropy analysis of second-harmonic generation collagen imaging of human pancreatic cancer.

Pancreatic ductal adenocarcinoma (PDAC) is one of the deadliest cancers with a minority (< 10%) of patients surviving five years past diagnosis. This could be improved with the development of new imaging modalities for early differentiation of benign and cancerous fibrosis. This study intends to explore the application of a two-photon microscopy technique known as second harmonic generation to PDAC using the 2D Wavelet Transform Modulus Maxima (WTMM) Anisotropy method to quantify collagen organization in fibrotic pancreatic tissue. Forty slides from PDAC patients were obtained and eight images were captured per each tissue category on each slide. Brownian surface motion and white noise images were generated for calibration and testing of a new variable binning approach to the 2D WTMM Anisotropy method. The variable binning method had greater resistance to wavelet scaling effects and white noise images were found to have the lowest anisotropy factor. Cancer and fibrosis had greater anisotropy factors (Fa) at small wavelet scales than normal and normal adjacent tissue. At a larger scale of 21 μm this relationship changed with normal tissue having a higher Fa than all other tissue groups. White noise is the best representative image for isotropy and the 2D WTMM anisotropy method is sensitive to changes induced in collagen by PDAC.

Elimination of Image Saturation Effects on Multifractal Statistics Using the 2D WTMM Method.

Imaging artifacts such as image saturation can restrict the computational analysis of medical images. Multifractal analyses are typically restricted to self-affine, everywhere singular, surfaces. Image saturation regions in these rough surfaces rob them of these core properties, and their exclusion decreases the statistical power of clinical analyses. By adapting the powerful 2D Wavelet Transform Modulus Maxima (WTMM) multifractal method, we developed a strategy where the image can be partitioned according to its localized response to saturated regions. By eliminating the contribution from those saturated regions to the partition function calculations, we show that the estimation of the multifractal statistics can be correctly calculated even with image saturation levels up to 20% (where 20% is the number of saturated pixels over the total number of pixels in the image).

Multiscale anisotropy analysis of second-harmonic generation collagen imaging of mouse skin.

.Significance: Morphological collagen signatures are important for tissue function, particularly in the tumor microenvironment. A single algorithmic framework with quantitative, multiscale morphological collagen feature extraction may further the use of collagen signatures in understanding fundamental tumor progression.Aim: A modification of the 2D wavelet transform modulus maxima (WTMM) anisotropy method was applied to both digitally simulated collagen fibers and second-harmonic-generation imaged collagen fibers of mouse skin to calculate a multiscale anisotropy factor to detect collagen fiber organization.Approach: The modified 2D WTMM anisotropy method was initially validated on synthetic calibration images to establish the robustness and sensitivity of the multiscale fiber organization tool. Upon validation, the algorithm was applied to collagen fiber organization in normal wild-type skin, melanoma stimulated skin, and integrin skin.Results: Normal wild-type skin collagen fibers have an increased anisotropy factor at all sizes scales. Interestingly, the multiscale anisotropy differences highlight important dissimilarities between collagen fiber organization in normal wild-type skin, melanoma stimulated, and integrin skin. At small scales ( to ), the integrin skin was vastly different than normal skin (), whereas the melanoma stimulated skin was vastly different than normal at large scales ( to , ).Conclusions: This objective computational collagen fiber organization algorithm is sensitive to collagen fiber organization across multiple scales for effective exploration of collagen morphological alterations associated with melanoma and the lack of integrin binding.

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.

Abstract P6-06-04: Exploratory computational longitudinal analysis of mammographic microenvironment disruption preceding breast tumorigenesis

Abstract Despite an aggressive emphasis on screening mammography, there has been no change in the number of initial metastatic breast cancer diagnosis since 1975. Even more alarming is the lack of effective screening to differentiate dormant cancerous pathology that may spontaneously ignite into an aggressive metastatic cancer. In this context, screening mammography is ineffective and biased towards a reactive diagnosis, when cancer is already present. We seek to develop a predictive algorithm that is sensitive to currently undetectable alterations of the microenvironment that precede tumorigenesis. Histological features such as tissue disruption and loss of tissue homeostasis, which are critical early indicators of breast cancer, unfortunately fall well outside the detection abilities of a trained radiologist or current computer-aided detection/diagnostic methods. In a recent publication using a powerful wavelet-based multifractal image analysis method, we extracted quantitative metrics for tissue disruption from standard mammography, and inferred a correlation with loss of tissue homeostasis. The 2D Wavelet-Transform Modulus Maxima (WTMM) method was used to quantify density fluctuations from mammographic breast tissue via a statistical roughness exponent called the Hurst exponent (H). The two mammographic tissue radiological states, i.e., fatty and dense, are quantitatively characterized with this Hurst exponent: H&amp;gt;1/2 (correlated spatial density fluctuations) for dense tissue; and H&amp;lt;1/2 (anti-correlated spatial density fluctuations) for fatty tissue. However, tissue regions that are neither dense nor fatty, with H~1/2 (uncorrelated spatial density fluctuations), were found preferably in patients with tumorous breasts vs. normal breasts (p-value = 0.0006) [Marin et al. Medical Physics, 2017]. The underlying physical processes associated with a H=1/2 signature are those of randomness and disorganization. Indeed, incoherent angular motion and the consequential adoption of randomized cellular motility are associated with malignant growth in breast tissue. We hypothesize that this H~1/2 signature accompanies or even preceeds tumorigenesis. In an exploratory analysis aimed at verifying this hypothesis, we studied, for 22 breast cancer patients, the progression of healthy breast tissue microenvironment towards a disrupted state by performing a computational analysis of longitudinal sequences of mammograms multiple years prior to, and leading to diagnostic. Preliminary results indicate a measurable change in mammographic density fluctuations (and therefore, potentially loss of tissue homeostasis), between 1 and 3 years prior to radiologic detection / diagnostic. An eventual larger-scale clinical validation of such an analysis could lead to the routine implementation of predictive algorithms based on the multifractal characterization of tissue disruption that precedes tumorigenesis. Such predictive diagnostics empowers both patient and clinician to collaboratively assess all ramification of the disease and select optimal therapeutics. Citation Format: Andre Khalil, Brian Toner. Exploratory computational longitudinal analysis of mammographic microenvironment disruption preceding breast tumorigenesis [abstract]. In: Proceedings of the 2019 San Antonio Breast Cancer Symposium; 2019 Dec 10-14; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2020;80(4 Suppl):Abstract nr P6-06-04.

Mammographic evidence of microenvironment changes in tumorous breasts.

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. 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. 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. 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.

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.

CHARACTERIZING COMPLEXITY IN SOLAR MAGNETOGRAM DATA USING A WAVELET-BASED SEGMENTATION METHOD

The multifractal nature of solar photospheric magnetic structures are studied using the 2D wavelet transform modulus maxima (WTMM) method. This relies on computing partition functions from the wavelet transform skeleton defined by the WTMM method. This skeleton provides an adaptive space-scale partition of the fractal distribution under study, from which one can extract the multifractal singularity spectrum. We describe the implementation of a multiscale image processing segmentation procedure based on the partitioning of the WT skeleton which allows the disentangling of the information concerning the multifractal properties of active regions from the surrounding quiet-Sun field. The quiet Sun exhibits a average H\"older exponent $\sim -0.75$, with observed multifractal properties due to the supergranular structure. On the other hand, active region multifractal spectra exhibit an average H\"older exponent $\sim 0.38$ similar to those found when studying experimental data from turbulent flows.

Image analysis of soft-tissue in-growth and attachment into highly porous alumina ceramic foam metals

The detailed quantitative characterization of soft-tissue in-growth into highly porous artificial implants is critical to understanding the biophysical processes that will lead to the best structural scaffolding construct. Previous studies have performed mechanical peel tests and mostly qualitative histological analyses of soft-tissue. The goal of this paper is to report the results obtained from applying two image analysis algorithms to quantify the morphological structure found in histological images of stained soft-tissue in-growth into alumina ceramic foam metal implants using a canine model. Three different pore sizes were used and three different post-operative time points were considered. Using the 2D Wavelet Transform Modulus Maxima method and 2D Fourier Transform analysis, a strong anisotropic signature (directional preference) is detected in early (4-week) histological samples. The direction of preference is towards the center of the implants. The strength of the anisotropy at later time points (8 and 16 weeks) becomes gradually weaker. Our interpretation is that after a short period of time, the main tissue growth activity has been concentrated on filling the artificial implant by growing towards its center. The weaker anisotropic signature found at later time points is interpreted as the tissue growth activity strengthening its structure by growing in more random directions.

Morphological Analysis of HiFeatures. II. Wavelet‐based Multifractal Formalism

The 2D wavelet transform modulus maxima method is used to obtain quantitative information on the fractal/multifractal nature and anisotropic structure of Galactic atomic hydrogen (H I) from the Canadian Galactic Plane Survey. Five mosaics were analyzed in the second quadrant of the Milky Way, corresponding to the Local, Perseus, and Outer spiral arms, as well as two regions located between these three spiral arms. A monofractal signature is found for all five mosaics. An anisotropic signature is detected: the rms roughness fluctuations of the mosaics in the latitude direction differ from those in the longitude direction. This anisotropy is scale-independent for the interarm regions while it is scale-dependent for the spiral arms. The longitudinal matter distribution of H I structure is similar for all five mosaics while the latitudinal distribution is smoother in the spiral arms. These results hold for all physical length scales studied, from ~2 pc in the Local arm to ~44 pc in the Outer arm. Several hypotheses are investigated to provide a physical explanation: the scale-height gradient, the density wave, star formation activity, photolevitation of dusty clouds, random motion of H I clouds, corrugation, and turbulence. The slopes of the power spectra for an increasing number of velocity channels were compared for 11 sections of the Local arm column density mosaic. All slopes are identical within the uncertainties (-3.0) and we do not detect for the Galactic plane the change in the power law index predicted by Lazarian and Pogosyan.

A wavelet-based method for multifractal image analysis. III. Applications to high-resolution satellite images of cloud structure

We apply the 2D wavelet transform modulus maxima (WTMM) method to high-resolution LANDSAT satellite images of cloudy scenes. The computation of the \(\) and D(h) multifractal spectra for both the optical depth and the radiance fields confirms the relevance of the multifractal description to account for the intermittent nature of marine stratocumulus clouds. When assisting the 2D WTMM method by the wavelet based deconvolution method designed to compute the self-similarity kernel, we show that our numerical tools are very efficient to disentangle the anisotropic texture induced by the presence of convective rolls from the background radiance fluctuations which are likely to display isotropic scale invariance. Moreover, this analysis reveals that with the available set of experimental data, there is no way to discriminate between various phenomenological cascade models recently proposed to account for intermittency and their log-normal approximations. When further investigating the “two-point” space-scale correlation functions, we bring definite proof of the existence of an underlying multiplicative structure from an “integral” coarsest scale which is given by the characteristic width of the convective patterns. We emphasize the log-normal random \(\)-cascade model on separable wavelet orthogonal basis introduced in paper II (N. Decoster, S.G. Roux, A. Arneodo, Eur. Phys. J. B 15, 739 (2000)), as a very attractive model (at least as compared to the models commonly used in the literature) of the cloud architecture. Finally, we comment on the multifractal properties of marine stratocumulus radiance fields comparatively to previous experimental analysis of velocity and temperature fluctuations in high Reynolds number turbulence.

A wavelet-based method for multifractal image analysis. II. Applications to synthetic multifractal rough surfaces

We apply the 2D wavelet transform modulus maxima (WTMM) method to synthetic random multifractal rough surfaces. We mainly focus on two specific models that are, a priori, reasonnable candidates to simulate cloud structure in paper III (S.G. Roux, A. Arneodo, N. Decoster, Eur. Phys. J. B 15, 765 (2000)). As originally proposed by Schertzer and Lovejoy, the first one consists in a simple power-law filtering (known in the mathematical literature as “fractional integration”) of singular cascade measures. The second one is the foremost attempt to generate log-infinitely divisible cascades on 2D orthogonal wavelet basis. We report numerical estimates of the \(\) and D(h) multifractal spectra which are in very good agreement with the theoretical predictions. We emphasize the 2D WTMM method as a very efficient tool to resolve multifractal scaling. But beyond the statistical information provided by the multifractal description, there is much more to learn from the arborescent structure of the wavelet transform skeleton of a multifractal rough surface. Various statistical quantities such as the self-similarity kernel and the space-scale correlation functions can be used to characterize very precisely the possible existence of an underlying multiplicative process. We elaborate theoretically and test numerically on various computer synthetized images that these statistical quantities can be directly extracted from the considered multifractal function using its WTMM skeleton with an arbitrary analyzing wavelets. This study provides algorithms that are readily applicable to experimental situations.