La matematica最新文献

A theoretical model of the human retina is simulated using two distinct vascular network geometries to predict the impact of heterogeneity in vascular network structure on retinal tissue oxygenation. Each vascular network is modeled as a combined heterogeneous representation of retinal arterioles and compartmental representation of capillaries, small venules, and large venules. A Green's function method is used to model oxygen transport in the arterioles, and a Krogh cylinder model is used in the capillaries and venules. Identical input arterial blood saturation (0.92), arteriolar pressure drop (16 mmHg), and arteriolar diameters by vessel order ( $117,73,44,32,\text{and}22\mu \text{m}$ ) are assumed for both networks. The model shows that 12% of the arteriolar tissue in Branch 1 has a PO₂ less than 25 mmHg, while only 1% of the arteriolar tissue in Branch 2 has a PO₂ less than 25 mmHg. However, downstream of the capillaries, Branch 2 was predicted to exhibit lower tissue PO₂ than Branch 1. The model also predicted increased oxygen extraction fraction as oxygen demand increased or capillary density decreased. Even with identical initial conditions for saturation, pressure drop, and diameter, variations in network geometry led to significantly different regions of low PO₂, indicating a wide range of potential oxygenation outcomes for individual patients. This study therefore demonstrates that regional heterogeneity in vessel branching architecture may significantly impact oxygen saturation and ultimately retinal ganglion cell functionality, motivating the need for creating patient-specific vascular networks.

Primary open angle glaucoma (POAG) is a leading cause of irreversible blindness with risk factors including elevated intraocular pressure (IOP), and both high and low blood pressure (BP). This study investigates the joint influence of IOP and BP on retinal hemodynamics, emphasizing venous circulation. A synthetic dataset comprising 2500 eyes with varied IOP [5-45] mmHg, systolic BP (SBP) [90-200] mmHg and diastolic BP (DBP) [40-120] mmHg was created. Mean pressure $\left(\bar{P}\right)$ , mean flow $\left(\bar{Q}\right)$ , and mean resistance $\left(\bar{R}\right)$ , were estimated using a validated mathematical model. The values of these hemodynamic output variables were then analyzed in relation to different values of IOP and mean arterial pressure (MAP; MAP = 1/3 SBP + 2/3 DBP). Clinical data from a population-based Greek study were similarly analyzed. Differences in the simulated hemodynamic output variables and clinical markers between healthy and POAG eyes were then measured. Synthetic dataset analysis revealed that $\bar{R}$ and $\bar{P}$ vary significantly depending on different IOP-MAP combinations. Notably, eyes with low MAP and high IOP demonstrated a drastic increase in $\bar{R}$ in the venules accompanied with a dramatic decrease in $\bar{P}$ in the central retinal vein (CRV). Clinical data indicated that venules in POAG eyes had significantly higher $\bar{R}$ than healthy eyes (p < 0.01), along with decreased $\bar{P}$ in the CRV of POAG eyes compared to healthy eyes (p = 0.01). The study highlights the increased susceptibility to venous collapse in POAG eyes and the importance of considering the venous side of retinal circulation in the combined impact of risk factors in POAG.

We study the puzzle graphs of hexagonal sliding puzzles of various shapes, and with various numbers of holes. The puzzle graph is a combinatorial model which captures the solvability and the complexity of sequential mechanical puzzles. Questions relating to the puzzle graph have been previously studied and resolved for the 15 Puzzle, which is the most famous-and unsolvable-square sliding puzzle of all time. It is known that for square puzzles such as the 15 Puzzle, solvability depends on a parity property that splits the puzzle graph into two components. In the case of hexagonal sliding puzzles, we get more interesting parity properties that depend on the shape of the boards and on the missing tiles or holes on the board. We show that for large-enough hexagonal, triangular, or parallelogram-shaped boards with hexagonal tiles, all puzzles with three or more holes are solvable. For puzzles with two or more holes, we give a solvability criterion involving both a parity property, and the placement of tiles in tight corners of the board. The puzzle graph is a discrete model for the configuration space of hard tiles (hexagons or squares) moving on different tessellation-based domains. Understanding the combinatorics of the puzzle graph could lead to understanding some aspects of the topology of these configuration spaces.

Patient-specific mathematical modeling combined with data analytics methods presents a promising approach for analyzing retinal hemodynamics. In this study, we build upon previous developments in retinal blood flow modeling, integrating clinical measurements and physiological insights to reconstruct parameters such as the central retinal artery velocity (CRA) Doppler profile and pressure wave. By leveraging personalized input data, including CRA velocity profile, systemic blood pressure, heart rate, and intraocular pressure, we evaluate the performance of our in silico approach. Our investigation highlights the significance of methodological considerations in combining automatic image processing and mathematical modeling, particularly concerning the selection of appropriate strategies and the inclusion of personalized supplementary data. Through extensive validation and comparison with prior works, we demonstrate the impact of different assessment methods on clinically meaningful quantities, particularly biomarkers related to blood flow. Furthermore, our study introduces a novel metric based on the Wasserstein distance for monitoring temporal changes in retinal blood flow dynamics, providing valuable insights into the evolution of vascular function. Overall, our findings underscore the importance of patient-specific input data, automatic image processing, and personalized mathematical modeling to ensure robust and clinically relevant outcomes in retinal vasculature analysis.