Shannon's entropy and fractal dimension provide an objective account of bone tissue organization during calvarial bone regeneration

Spatial mapping of collagen content and structure in human intervertebral disk degeneration

Collagen plays a key structural role in both the annulus fibrosus (AF) and nucleus pulposus (NP) of intervertebral disks (IVDs). Changes in collagen content with degeneration suggest a shift from collagen type II to type I within the NP, and the activation of pro-inflammatory factors is indicative of fibrosis throughout. While IVD degeneration is considered a fibrotic process, an increase in collagen content with degeneration, reflective of fibrosis, has not been demonstrated. Additionally, changes in collagen content and structure in human IVDs with degeneration have not been characterized with high spatial resolution. The collagen content of 23 human lumbar L2/3 or L3/4 IVDs was quantified using second harmonic generation imaging (SHG) and multiple image processing algorithms, and these parameters were correlated with the Rutges histological degeneration grade. In the NP, SHG intensity increased with degeneration grade, suggesting fibrotic collagen deposition. In the AF, the entropy of SHG intensity was reduced with degeneration indicating increased collagen uniformity and suggesting less-organized lamellar structure. Collagen orientation entropy decreased throughout most IVD regions with increasing degeneration grade, further supporting a loss in collagen structural complexity. Overall, SHG imaging enabled visualization and quantification of IVD collagen content and organization with degeneration. There was an observed shift from an initially complex structure to more uniform structure with loss of microstructural elements and increased NP collagen polarity, suggesting fibrotic remodeling.

Toward a Pragmatic Theory for Managing Nescience

Aristotle’s dictum scio nescio (I know that I don’t know) may serve as a source of enhanced performance for organizations. Awareness of nescience sets the direction for further inquiry, as managers tend to move in the direction that they believe will reduce nescience most. However, nescience is difficult to quantify, so, to date, managers have primarily relied on intuition. This paper introduces a theoretical framework for managing nescience that is based on information theory. This framework is tested in three exploratory empirical studies that take place in highly contrasting settings: semiconductor manufacturing, medical diagnostics and social media analytics. All three studies demonstrate that metrics related to information entropy can be used to quantify nescience. However, practitioners value the framework and its metrics more highly in the settings where the quality of or access to information drives successful product development. The problems encountered in these settings tended to be well-structured, or they were converted from being ill-structured to being well-structured. Further study of more highly contrasting practical settings will be required to determine whether frameworks based on information theory can serve as foundations for a broadly based, pragmatic theory for managing nescience.

Routes to fractality and entropy in Liesegang systems

Liesegang bands are formed when solutions of co-precipitate ions interdiffuse in a 1D gel matrix. In a recent study [R. F. Sultan, Acta. Mech. Sin. 27, 119 (2011)], Liesegang patterns have been characterized as fractal structures. In addition to experimentally obtained patterns, geometric Liesegang patterns were constructed in conformity with the well-known empirical laws. Both mathematical fractal dimensions and box count dimensions for images of PbF2 and PbI2 Liesegang patterns have been calculated. Liesegang patterns can also be described by the entropy state function, and categorized as more or less ordered structures. We revisit the relation between entropy and fractal dimension, and apply it to simulated geometrical Liesegang patterns. We have resort to three different routes for the estimation of the entropy of a Liesegang pattern. The HarFA software enabled the calculation of the Hausdorff dimension and the topological entropy, then the information dimension and the Shannon entropy. In a third pathway, analytical calculations were carried out by estimating the probability of occurrence of a fractal element or coverage. The product of Shannon entropy and Boltzmann constant yields the thermodynamic entropy. The values for PbF2 and PbI2 Liesegang patterns attained the order of magnitude of the reported Third Law entropies, but yet remained lower, in conformity with the more ordered Liesegang structures.

Fractal dimension of chromatin: potential molecular diagnostic applications for cancer prognosis

Fractal characteristics of chromatin, revealed by light or electron microscopy, have been reported during the last 20 years. Fractal features can easily be estimated in digitalized microscopic images and are helpful for diagnosis and prognosis of neoplasias. During carcinogenesis and tumor progression, an increase of the fractal dimension (FD) of stained nuclei has been shown in intraepithelial lesions of the uterine cervix and the anus, oral squamous cell carcinomas or adenocarcinomas of the pancreas. Furthermore, an increased FD of chromatin is an unfavorable prognostic factor in squamous cell carcinomas of the oral cavity and the larynx, melanomas and multiple myelomas. High goodness-of-fit of the regression line of the FD is a favorable prognostic factor in acute leukemias and multiple myelomas. The nucleus has fractal and power-law organization in several different levels, which might in part be interrelated. Some possible relations between modifications of the chromatin organization during carcinogenesis and tumor progression and an increase of the FD of stained chromatin are suggested. Furthermore, increased complexity of the chromatin structure, loss of heterochromatin and a less-perfect self-organization of the nucleus in aggressive neoplasias are discussed.

Fractal dimension and Shannon's entropy analyses of the architectural complexity caused by the inflammatory reactions induced by highly crystalline poly(vinyl alcohol) microspheres implanted in subcutaneous tissues of the Wistar rats

The results of the histopathological analyses after the implantation of highly crystalline PVA microspheres in subcutaneous tissues of Wistar rats are here in reported. Three different groups of PVA microparticles were systematically studied: highly crystalline, amorphous, and commercial ones. In addition to these experiments, complementary analyses of architectural complexity were performed using fractal dimension (FD), and Shannon's entropy (SE) concepts. The highly crystalline microspheres induced inflammatory reactions similar to the ones observed for the commercial ones, while the inflammatory reactions caused by the amorphous ones were less intense. Statistical analyses of the subcutaneous tissues of Wistar rats implanted with the highly crystalline microspheres resulted in FD and SE values significantly higher than the statistical parameters observed for the amorphous ones. The FD and SE parameters obtained for the subcutaneous tissues of Wistar rats implanted with crystalline and commercial microparticles were statistically similar. Briefly, the results indicated that the new highly crystalline microspheres had biocompatible behavior comparable to the commercial ones. In addition, statistical tools such as FD and SE analyses when combined with histopathological analyses can be useful tools to investigate the architectural complexity tissues caused by complex inflammatory reactions.

Assessment of the protective effects of oral tocotrienols in arginine chronic-like pancreatitis

Tocotrienols exhibit anti-inflammatory properties over macrophages and promote cytotoxicity in activated pancreatic stellate cells, suggesting that they may limit chronic pancreatitis progression. We aimed to quantitate the effect of oral tocotrienols on a rat model of chronic pancreatic injury. Chronic-like pancreatitis was induced by repeated arginine pancreatitis. Palm oil tocotrienol-rich fraction (TRF) was given by gavage before and after pancreatitis inductions. Amylase and hydroxyproline were determined in pancreatic homogenates; collagen, fibronectin, α-smooth muscle actin (SMA), glial fibrillary acidic protein (GFAP), and phosphorylated Smad3 were assessed by Western blotting. Transforming growth factor (TGF)-β1 was measured in plasma. Morphological assessment included light microscopy, fibrosis area fraction, and collagen network fractal analysis. Arginine pancreatitis induced pancreatic atrophy and increased hydroxyproline that ameliorated after TRF. Arginine increased TGF-β1 (185 ± 40 vs. 15 ± 2 ng/ml; P <0.01) that was blunted by TRF (53 ± 19; P < 0.01). TRF reduced protease and Smad3 activation, collagen, and fibronectin. α-SMA increased and GFAP diminished in arginine pancreatitis, consistent with long-term stellate cell activation, and TRF reverted these changes to basal. Arginine pancreatitis increased fibrosis area fraction (4.5 ± 0.3% vs. 0.2 ± 0.2%), collagen network complexity (fractal dimension 1.52 ± 0.03 vs. 1.42 ± 0.01; P < 0.001), and inhomogeneity (lacunarity 0.63 ± 0.03 vs. 0.40 ± 0.02; P < 0.001), which were all reduced by TRF (1.3 ± 0.4%, 1.43 ± 0.02%, and 0.51 ± 0.03%, respectively; P < 0.01). Best correlation coefficients were obtained when comparing fibrosis area fraction with lacunarity (r = 0.88) and both parameters with pancreatic weight (r = -0.91 and -0.79, respectively). TRF administered only before pancreatitis best, but not fully, recapitulated the beneficial effects of TRF. Tocotrienols improve quantitative measures of chronic pancreatic damage. They may be of benefit in human chronic pancreatitis.

Fractal characteristics of May-Grünwald-Giemsa stained chromatin are independent prognostic factors for survival in multiple myeloma

BACKGROUND

The use of computerized image analysis for the study of nuclear texture features has provided important prognostic information for several neoplasias. Recently fractal characteristics of the chromatin structure in routinely stained smears have shown to be independent prognostic factors in acute leukemia. In the present study we investigated the influence of the fractal dimension (FD) of chromatin on survival of patients with multiple myeloma.

METHODOLOGY

We analyzed 67 newly diagnosed patients from our Institution treated in the Brazilian Multiple Myeloma Study Group. Diagnostic work-up consisted of peripheral blood counts, bone marrow cytology, bone radiograms, serum biochemistry and cytogenetics. The International Staging System (ISS) was used. In every patient, at least 40 digital nuclear images from diagnostic May-Grünwald-Giemsa stained bone marrow smears were acquired and transformed into pseudo-3D images. FD was determined by the Minkowski-Bouligand method extended to three dimensions. Goodness-of-fit of FD was estimated by the R(2) values in the log-log plots. The influence of diagnostic features on overall survival was analyzed in Cox regressions. Patients that underwent autologous bone marrow transplantation were censored at the day of transplantation.

PRINCIPAL FINDINGS

Median age was 56 years. According to ISS, 14% of the patients were stage I, 39% were stage II and 47% were stage III. Additional features of a bad prognosis were observed in 46% of the cases. When stratifying for ISS, both FD and its goodness-of-fit were significant prognostic factors in univariate analyses. Patients with higher FD values or lower goodness-of-fit showed a worse outcome. In the multivariate Cox-regression, FD, R(2), and ISS stage entered the final model, which showed to be stable in a bootstrap resampling study.

CONCLUSIONS

Fractal characteristics of the chromatin texture in routine cytological preparations revealed relevant prognostic information in patients with multiple myeloma.