Estimating fractal dimension with fractal interpolation function models

Validation of an Artificial Intelligence-Based Ultrasound Imaging System for Quantifying Muscle Architecture Parameters of the Rectus Femoris in Disease-Related Malnutrition (DRM)

(1) Background: The aim was to validate an AI-based system compared to the classic method of reading ultrasound images of the rectus femur (RF) muscle in a real cohort of patients with disease-related malnutrition. (2) Methods: One hundred adult patients with DRM aged 18 to 85 years were enrolled. The risk of DRM was assessed by the Global Leadership Initiative on Malnutrition (GLIM). The variation, reproducibility, and reliability of measurements for the RF subcutaneous fat thickness (SFT), muscle thickness (MT), and cross-sectional area (CSA), were measured conventionally with the incorporated tools of a portable ultrasound imaging device (method A) and compared with the automated quantification of the ultrasound imaging system (method B). (3) Results: Measurements obtained using method A (i.e., conventionally) and method B (i.e., raw images analyzed by AI), showed similar values with no significant differences in absolute values and coefficients of variation, 58.39-57.68% for SFT, 30.50-28.36% for MT, and 36.50-36.91% for CSA, respectively. The Intraclass Correlation Coefficient (ICC) for reliability and consistency analysis between methods A and B showed correlations of 0.912 and 95% CI [0.872-0.940] for SFT, 0.960 and 95% CI [0.941-0.973] for MT, and 0.995 and 95% CI [0.993-0.997] for CSA; the Bland-Altman Analysis shows that the spread of points is quite uniform around the bias lines with no evidence of strong bias for any variable. (4) Conclusions: The study demonstrated the consistency and reliability of this new automatic system based on machine learning and AI for the quantification of ultrasound imaging of the muscle architecture parameters of the rectus femoris muscle compared with the conventional method of measurement.

The fractal geometry of Hartree-Fock

The Hartree-Fock method is an important approximation for the ground-state electronic wave function of atoms and molecules so that its usage is widespread in computational chemistry and physics. The Hartree-Fock method is an iterative procedure in which the electronic wave functions of the occupied orbitals are determined. The set of functions found in one step builds the basis for the next iteration step. In this work, we interpret the Hartree-Fock method as a dynamical system since dynamical systems are iterations where iteration steps represent the time development of the system, as encountered in the theory of fractals. The focus is put on the convergence behavior of the dynamical system as a function of a suitable control parameter. In our case, a complex parameter λ controls the strength of the electron-electron interaction. An investigation of the convergence behavior depending on the parameter λ is performed for helium, neon, and argon. We observe fractal structures in the complex λ-plane, which resemble the well-known Mandelbrot set, determine their fractal dimension, and find that with increasing nuclear charge, the fragmentation increases as well.

A Review of the Quantification and Classification of Pigmented Skin Lesions: From Dedicated to Hand-Held Devices

In recent years, the incidence of skin cancer cases has risen, worldwide, mainly due to the prolonged exposure to harmful ultraviolet radiation. Concurrently, the computer-assisted medical diagnosis of skin cancer has undergone major advances, through an improvement in the instrument and detection technology, and the development of algorithms to process the information. Moreover, because there has been an increased need to store medical data, for monitoring, comparative and assisted-learning purposes, algorithms for data processing and storage have also become more efficient in handling the increase of data. In addition, the potential use of common mobile devices to register high-resolution images of skin lesions has also fueled the need to create real-time processing algorithms that may provide a likelihood for the development of malignancy. This last possibility allows even non-specialists to monitor and follow-up suspected skin cancer cases. In this review, we present the major steps in the pre-processing, processing and post-processing of skin lesion images, with a particular emphasis on the quantification and classification of pigmented skin lesions. We further review and outline the future challenges for the creation of minimum-feature, automated and real-time algorithms for the detection of skin cancer from images acquired via common mobile devices.

Quantifying tumour heterogeneity with CT

Heterogeneity is a key feature of malignancy associated with adverse tumour biology. Quantifying heterogeneity could provide a useful non-invasive imaging biomarker. Heterogeneity on computed tomography (CT) can be quantified using texture analysis which extracts spatial information from CT images (unenhanced, contrast-enhanced and derived images such as CT perfusion) that may not be perceptible to the naked eye. The main components of texture analysis can be categorized into image transformation and quantification. Image transformation filters the conventional image into its basic components (spatial, frequency, etc.) to produce derived subimages. Texture quantification techniques include structural-, model- (fractal dimensions), statistical- and frequency-based methods. The underlying tumour biology that CT texture analysis may reflect includes (but is not limited to) tumour hypoxia and angiogenesis. Emerging studies show that CT texture analysis has the potential to be a useful adjunct in clinical oncologic imaging, providing important information about tumour characterization, prognosis and treatment prediction and response.

Scale-free brain-wave music from simultaneously EEG and fMRI recordings

In the past years, a few methods have been developed to translate human EEG to music. In 2009, PloS One 4 e5915, we developed a method to generate scale-free brainwave music where the amplitude of EEG was translated to music pitch according to the power law followed by both of them, the period of an EEG waveform is translated directly to the duration of a note, and the logarithm of the average power change of EEG is translated to music intensity according to the Fechner's law. In this work, we proposed to adopt simultaneously-recorded fMRI signal to control the intensity of the EEG music, thus an EEG-fMRI music is generated by combining two different and simultaneous brain signals. And most importantly, this approach further realized power law for music intensity as fMRI signal follows it. Thus the EEG-fMRI music makes a step ahead in reflecting the physiological process of the scale-free brain.

A fully automated scheme for mammographic segmentation and classification based on breast density and asymmetry

This paper presents a fully automated segmentation and classification scheme for mammograms, based on breast density estimation and detection of asymmetry. First, image preprocessing and segmentation techniques are applied, including a breast boundary extraction algorithm and an improved version of a pectoral muscle segmentation scheme. Features for breast density categorization are extracted, including a new fractal dimension-related feature, and support vector machines (SVMs) are employed for classification, achieving accuracy of up to 85.7%. Most of these properties are used to extract a new set of statistical features for each breast; the differences among these feature values from the two images of each pair of mammograms are used to detect breast asymmetry, using an one-class SVM classifier, which resulted in a success rate of 84.47%. This composite methodology has been applied to the miniMIAS database, consisting of 322 (MLO) mammograms -including 15 asymmetric pairs of images-, obtained via a (noisy) digitization procedure. The results were evaluated by expert radiologists and are very promising, showing equal or higher success rates compared to other related works, despite the fact that some of them used only selected portions of this specific mammographic database. In contrast, our methodology is applied to the complete miniMIAS database and it exhibits the reliability that is normally required for clinical use in CAD systems.

Signal attenuation and box-counting fractal analysis of optical coherence tomography images of arterial tissue

The sensitivity of optical coherence tomography images to sample morphology is tested by two methods. The first method estimates the attenuation of the OCT signal from various regions of the probed tissue. The second method uses a box-counting algorithm to calculate the fractal dimensions in the regions of interest identified in the images. Although both the attenuation coefficient as well as the fractal dimension correlate very well with the anatomical features of the probed samples; the attenuation method provides a better sensitivity. Two types of samples are used in this study: segments of arteries collected from atherosclerosis-prone Watanabe rabbits (WHHL-MI) and healthy segments of porcine coronary arteries.

Texture analysis of aggressive and nonaggressive lung tumor CE CT images

This paper presents the potential for fractal analysis of time sequence contrast-enhanced (CE) computed tomography (CT) images to differentiate between aggressive and nonaggressive malignant lung tumors (i.e., high and low metabolic tumors). The aim is to enhance CT tumor staging prediction accuracy through identifying malignant aggressiveness of lung tumors. As branching of blood vessels can be considered a fractal process, the research examines vascularized tumor regions that exhibit strong fractal characteristics. The analysis is performed after injecting 15 patients with a contrast agent and transforming at least 11 time sequence CE CT images from each patient to the fractal dimension and determining corresponding lacunarity. The fractal texture features were averaged over the tumor region and quantitative classification showed up to 83.3% accuracy in distinction between advanced (aggressive) and early-stage (nonaggressive) malignant tumors. Also, it showed strong correlation with corresponding lung tumor stage and standardized tumor uptake value of fluorodeoxyglucose as determined by positron emission tomography. These results indicate that fractal analysis of time sequence CE CT images of malignant lung tumors could provide additional information about likely tumor aggression that could potentially impact on clinical management decisions in choosing the appropriate treatment procedure.

Fractal analysis of mammographic parenchymal patterns in breast cancer risk assessment

RATIONALE AND OBJECTIVES

To evaluate fractal-based computerized image analyses of mammographic parenchymal patterns in the task of differentiating between women at high risk and women at low risk for developing breast cancer.

MATERIALS AND METHODS

The fractal-based texture analyses are based on a box-counting method and a Minkowski dimension, and were performed within the parenchymal regions of normal mammograms. Four approaches were evaluated: 1) a conventional box-counting method, 2) a modified box-counting technique using linear discriminant analysis (LDA), 3) a global Minkowski dimension, and 4) a modified Minkowski technique using LDA. These fractal based texture features were extracted from regions of interest to assess the mammographic parenchymal patterns of the images. Receiver operating characteristic analysis was used to evaluate the performance of these features in the task of differentiating between the two groups of women.

RESULTS

Receiver operating characteristic analysis yielded an A(z) value of 0.74 based on the conventional box-counting technique and an A(z) value of 0.84 based on the global Minkowski dimension in the task of distinguishing between the two groups. By using LDA to assess the characteristics of mammograms, A(z) values of 0.90 and 0.93 were obtained in differentiating the two groups, for the modified box-counting and Minkowski techniques, respectively. Statistically significant improvement was achieved (P < .05) with the new techniques compared to the conventional fractal analysis methods. A simulation study, which used the slope and intercept extracted from the least square fit of the experimental data with the LDA approaches, yielded A(z) values similar to those obtained with the conventional approaches in the task of differentiating between the two groups.

CONCLUSIONS

The proposed LDA approach improved significantly the separation between the two groups based on experimental data. Because this approach was used as a linear classifier rather than as a regression function, it combined the fractal analysis with the knowledge of the high- and low-risk patterns, and thus better characterized the multifractal nature of the parenchymal patterns. We believe that the proposed analyses based on the LDA technique to characterize mammographic parenchymal patterns may potentially yield radiographic markers for assessing breast cancer risk.

Automated analysis of contrast enhancement in breast MRI lesions using mean shift clustering for ROI selection

PURPOSE

To evaluate a new method for automated determination of a region of interest (ROI) for the analysis of contrast enhancement in breast MRI.

MATERIALS AND METHODS

Mean shift multidimensional clustering (MS-MDC) was employed to divide 92 lesions into several spatially contiguous clusters each, based on multiple enhancement parameters. The ROIs were defined as the clusters with the highest probability of malignancy. The performance of enhancement analysis within these ROIs was estimated using the area under the receiver operator characteristic curve (AUC), and compared against a radiologist's final assessment and a classifier using histogram analysis (HA). For HA, the first, second, and third quartiles were evaluated.

RESULTS

MS-MDC resulted in AUC = 0.88 with a 95% confidence interval (CI) of 0.81-0.95. The AUC for the radiologist's assessment was 0.93 (95%CI = 0.87-0.97). Best HA performance was found using the first quartile, with AUC = 0.79 (95%CI = 0.69-0.88). There was no significant difference between MS-MDC and the radiologist (P = 0.40). The improvement of MS-MDC over HA was significant (P = 0.018).

CONCLUSION

Mean shift clustering followed by automated selection of the most suspicious cluster resulted in accurate ROIs in breast MRI lesions.

Anatomy and flow in normal and ischemic microvasculature based on a novel temporal fractal dimension analysis algorithm using contrast enhanced ultrasound

Strategies for improvement of blood flow by promoting new vessel growth in ischemic tissue are being developed. Recently, contrast-enhanced ultrasound (CEU) imaging has been used to assess tissue perfusion in models of ischemia-related angiogenesis, growth-factor mediated angiogenesis, and tumor angiogenesis. In these studies, microvascular flow is measured in order to assess the total impact of adaptations at different vascular levels. High-resolution methods for imaging larger vessels have been developed in order to derive "angiograms" of arteries, veins, and medium to large microvessels. We describe a novel method of vascular bed (microvessel and arterial) characterization of vessel anatomy and flow simultaneously, using serial measurement of the fractal dimension (FD) of a temporal sequence of CEU images. This method is proposed as an experimental methodology to distinguish ischemic from nonischemic tissue. Moreover, an improved approach for extracting the FD unique to this application is introduced.

Computerized texture analysis of mammographic parenchymal patterns of digitized mammograms

RATIONALE AND OBJECTIVES

Mammographic density and parenchymal patterns have been shown to be related to the risk of developing breast cancer. Thus, computerized texture analysis of breast parenchymal patterns on mammograms may be useful in assessing breast cancer risk.

MATERIALS AND METHODS

A comparative evaluation was conducted of various computer-extracted texture features of mammographic parenchymal patterns of women with BRCA1/BRCA2 gene mutations and those of women at low risk of developing breast cancer. Mammograms from 172 subjects (30 women with either the BRCA1 or BRCA2 gene mutation and 142 low-risk women) were analyzed. Computerized texture features were extracted from regions-of-interest to assess the mammographic parenchymal patterns in the images. Receiver operating characteristic analysis was used to assess the performance of these features in the task of distinguishing between the two groups of women.

RESULTS

Quantitative texture analysis on digitized mammograms demonstrated that gene-mutation carriers and low-risk women have different mammographic parenchymal patterns. Gene-mutation carriers presented with parenchymal patterns that were denser, coarser, and lower in contrast than those of the low-risk group. For the gene-mutation carriers, their mammographic patterns appear to contain less high-frequency component as indicated by higher coarseness values, lower fractal dimensions, and smaller edge gradients, which yielded corresponding A(z) values of 0.79, 0.84, and 0.78, respectively, in the task of distinguishing between gene-mutation carriers and the low-risk group with the entire dataset. The contrast measure calculated from co-occurrence matrix method, which describes local image variation, yielded an A(z) value of 0.86 in distinguishing between the two groups of women.

CONCLUSION

Computerized texture analysis of mammograms provides radiographic descriptors of mammographic parenchymal patterns. The computer-extracted features may be useful for identifying women at high risk for breast cancer and for monitoring the treatment of breast cancer patients.

Fractal dynamics in the differential diagnosis of thyroideal follicular neoplasms

In this study, we investigated the sensitivity, specificity, and predictive values of morphometric parameters of thyroideal follicular neoplasms based on concepts of fractal geometry. Thirty-seven follicular adenomas and 36 well-differentiated follicular carcinomas were assessed morphometrically. The nuclear area, nuclear area fraction, nuclear regularity factor, nuclear elongation factor, and slope setting (representing the ratio between the nuclear perimeter and nuclear regularity factor) were subjected to fractal dimensions analysis. By univariate analysis, the nuclear area, nuclear area fraction, nuclear regularity factor and slope values discriminate between adenomas and carcinomas. By multivariate analysis, the nuclear area, nuclear area fraction and slope values possess significant discriminatory powers in distinguishing between adenomas and carcinomas. Incorporating the nuclear area, nuclear area fraction, and slope values leads to a discriminatory power with 92% specificity and 83% sensitivity. The reciprocal relationships between the nuclear area, nuclear perimeter, and nuclear regularity factor of the cells of thyroideal adenomas and carcinomas may be expressed by fractal dimensions. Analysis limited to one parameter provides incomplete data. Expressing variations of the nuclear perimeter as a function of the nuclear regularity factor, the slope values constitute an independent attribute that significantly differentiates thyroideal adenomas from carcinomas.

Mammogram synthesis using a 3D simulation. II. Evaluation of synthetic mammogram texture

We have evaluated a method for synthesizing mammograms by comparing the texture of clinical and synthetic mammograms. The synthesis algorithm is based upon simulations of breast tissue and the mammographic imaging process. Mammogram texture was synthesized by projections of simulated adipose tissue compartments. It was hypothesized that the synthetic and clinical texture have similar properties, assuming that the mammogram texture reflects the 3D tissue distribution. The size of the projected compartments was computed by mathematical morphology. The texture energy and fractal dimension were also computed and analyzed in terms of the distribution of texture features within four different tissue regions in clinical and synthetic mammograms. Comparison of the cumulative distributions of the mean features computed from 95 mammograms showed that the synthetic images simulate the mean features of the texture of clinical mammograms. Correlation of clinical and synthetic texture feature histograms, averaged over all images, showed that the synthetic images can simulate the range of features seen over a large group of mammograms. The best agreement with clinical texture was achieved for simulated compartments with radii of 4-13.3 mm in predominantly adipose tissue regions, and radii of 2.7-5.33 and 1.3-2.7 mm in retroareolar and dense fibroglandular tissue regions, respectively.

Three-dimensional tumor perfusion reconstruction using fractal interpolation functions

It has been shown that the perfusion of blood in tumor tissue can be approximated using the relative perfusion index determined from dynamic contrast-enhanced magnetic resonance imaging (DE-MRI) of the tumor blood pool. Also, it was concluded in a previous report that the blood perfusion in a two-dimensional (2-D) tumor vessel network has a fractal structure and that the evolution of the perfusion front can be characterized using invasion percolation. In this paper, the three-dimensional (3-D) tumor perfusion is reconstructed from the 2-D slices using the method of fractal interpolation functions (FIF), i.e., the piecewise self-affine fractal interpolation model (PSAFIM) and the piecewise hidden variable fractal interpolation model (PHVFIM). The fractal models are compared to classical interpolation techniques (linear, spline, polynomial) by means of determining the 2-D fractal dimension of the reconstructed slices. Using FIFs instead of classical interpolation techniques better conserves the fractal-like structure of the perfusion data. Among the two FIF methods, PHVFIM conserves the 3-D fractality better due to the cross correlation that exists between the data in the 2-D slices and the data along the reconstructed direction. The 3-D structures resulting from PHVFIM have a fractal dimension within 3%-5% of the one reported in literature for 3-D percolation. It is, thus, concluded that the reconstructed 3-D perfusion has a percolation-like scaling. As the perfusion term from bio-heat equation is possibly better described by reconstruction via fractal interpolation, a more suitable computation of the temperature field induced during hyperthermia treatments is expected.

Discrimination of MR images of breast masses with fractal-interpolation function models

RATIONALE AND OBJECTIVES

The authors evaluated the feasibility of using statistical fractal-dimension features to improve discrimination between benign and malignant breast masses at magnetic resonance (MR) imaging.

MATERIALS AND METHODS

The study evaluated MR images of 32 malignant and 20 benign breast masses from archived data at the University of Pennsylvania Medical Center. The test set included four cases that were difficult to evaluate on the basis of border characteristics. All diagnoses had been confirmed at excisional biopsy. The fractal-dimension feature was computed as the mean of a sample space of fractal-dimension estimates derived from fractal interpolation function models. To evaluate the performance of the fractal-dimension feature, the classification effectiveness of five expert-observer architectural features was compared with that of the fractal dimension combined with four expert-observer features. Feature sets were evaluated with receiver operating characteristic analysis. Discrimination analysis used artificial neural networks and logistic regression. Robustness of the fractal-dimension feature was evaluated by determining changes in discrimination when the algorithm parameters were perturbed.

RESULTS

The combination of fractal-dimension and expert-observer features provided a statistically significant improvement in discrimination over that achieved with expert-observer features alone. Perturbing selected parameters in the fractal-dimension algorithm had little effect on discrimination.

CONCLUSION

A statistical fractal-dimension feature appears to be useful in distinguishing MR images of benign and malignant breast masses in cases where expert radiologists may have difficulty. The statistical approach to estimating the fractal dimension appears to be more robust than other fractal measurements on data-limited medical images.