Conley-Morse-Forman theory for generalized combinatorial multivector fields on finite topological spaces.

We generalize and extend the Conley-Morse-Forman theory for combinatorial multivector fields introduced in Mrozek (Found Comput Math 17(6):1585-1633, 2017). The generalization is threefold. First, we drop the restraining assumption in Mrozek (Found Comput Math 17(6):1585-1633, 2017) that every multivector must have a unique maximal element. Second, we define the dynamical system induced by the multivector field in a less restrictive way. Finally, we also change the setting from Lefschetz complexes to finite topological spaces. Formally, the new setting is more general, because every Lefschetz complex is a finite topological space, but the main reason for switching to finite topologcial spaces is because the latter better explain some peculiarities of combinatorial topological dynamics. We define isolated invariant sets, isolating neighborhoods, Conley index and Morse decompositions. We also establish the additivity property of the Conley index and the Morse inequalities.

Conley Index Approach to Sampled Dynamics.

The topological method for the reconstruction of dynamics from time series [K. Mischaikow et al., Phys. Rev. Lett., 82 (1999), pp. 1144-1147] is reshaped to improve its range of applicability, particularly in the presence of sparse data and strong expansion. The improvement is based on a multivalued map representation of the data. However, unlike the previous approach, it is not required that the representation has a continuous selector. Instead of a selector, a recently developed new version of Conley index theory for multivalued maps [B. Batko, SIAM J. Appl. Dyn. Syst., 16 (2017), pp. 1587-1617; B. Batko and M. Mrozek, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 1143-1162] is used in computations. The existence of a continuous, single valued generator of the relevant dynamics is guaranteed in the vicinity of the graph of the multivalued map constructed from data. Some numerical examples based on time series derived from the iteration of Hénon-type maps are presented.

Sensitivity and Specificity of Fractal Analysis to Distinguish Between Healthy and Pathologic Rectal Mucosa Microvasculature Seen During Colonoscopy.

PURPOSE: Conventional endoscopy is limited by human capability to recognize and to differentiate pathology. Fractal analysis of blood vessels has been used in other organs, such as the retina, but never before to supplement colonoscopy. The aim of this study was to assess whether it is possible to differentiate between healthy and pathologic rectal mucosa using fractal analysis of the mucosal microvascular architecture seen during colonoscopic examination (Supplemental Digital Content 1, http://links.lww.com/SLE/A145). METHODS: A total of 300 consecutive patients, 133 females and 167 males with a mean age of 49.1 (±11.3) years, undergoing endoscopy were included in the prospective cohort study. Colonoscopy of the sigmorectal region was performed, and then analyzed with computer-assisted image fractal analysis. RESULTS: Fractal analysis of mucosal vasculature allowed for differentiation between healthy and pathologic rectal mucosa, as well as different pathologies (P<0.0001). The sensitivity of fractal analysis to diagnose rectal neoplasia was 92.8% to 96.4%, while the specificity was 91.9% to 98.5% depending on the fractal parameter. The sensitivity of fractal analysis to diagnose rectal colitis was 84.2% to 92.1%, while the specificity was 95.0% to 96.0%, depending on the fractal parameter. CONCLUSIONS: Computer-assisted fractal analysis allows for differentiation between healthy and pathologic rectal mucosa, as well as between different mucosal pathologies, seen during colonoscopy. Fractal analysis improves the sensitivity and specificity of colonoscopy to aid in the diagnosis of neoplasia or colitis, and should be included in the screening and surveillance of these pathologies.

Fractal dimensions of skin microcirculation flow in subjects with familial predisposition or newly diagnosed hypertension.

BACKGROUND: Fractal analysis has been shown to be capable of characterizing irregular time series generated in non-linear systems. Fluctuations in skin flow signals have a fractal nature, but to date there has been no analysis of subjects with hypertension. The aim of this study is to assess the fractal dimensions of skin microcirculation flows in subjects with a familial predisposition or newly diagnosed hypertension. METHODS: A four-minutes rest flow (RF), minimal flow (BZ) during three-minutes ischemia and eight-minutes heat flow (HF) were recorded (using laser Doppler flowmetry) in patients with untreated hypertension, and in normotensives with no [NT(-)] or with a familial predisposition to hypertension [NT(+)]. Average one-minute surface areas under the curve of flow records and box dimensions (D) were calculated. Anova Kruskall-Wallis, c2 statistic and multivariate reverse regression analysis were used for calculation. RESULTS: We studied 70 people (average age 36.1 ± 10.3 years, 39 men). Hypertensives (n = 31) had significantly higher values of both clinical blood pressure and 24-hour ambulatory blood pressure, body mass index, glucose, triglycerides and insulin than the NT(-), (n = 17) and NT(+), (n = 22) groups. Mean values of flows and surface area under the curve of RF, BZ, HF records, D RF and D HF were comparable in studied groups, but D BZ differed (1.13 ± ± 0.05 vs 1.15 ± 0.05 vs 1.11 ± 0.05, respectively; p = 0.04). A family history of hypertension, insulin level and variability of 24-hour diastolic blood pressure were significant predictors of D BZ lower values in the multiple regression model. CONCLUSIONS: Subjects with a familial predisposition to hypertension reveal altered homeodynamics of microvascular flow, with diminished chaotic ischemic flow.