Investigation into the diversity in the fractal dimensions of arterioles and venules in a microvascular network – A quantitative analysis

Visualizing the spatiotemporal pattern of yolk sac membrane vascular network by enhanced local fractal analysis

OBJECTIVE

To establish methods for providing a comprehensive and detailed description of the spatial distribution of the vascular networks, and to reveal the spatiotemporal pattern of the yolk sac membrane vascular network during the angiogenic procedure.

METHODS

Addressing the limitations in the conventional local fractal analysis, an improved approach, named scanning average local fractal dimension, was proposed. This method was conducted on 6 high-resolution vascular images of the yolk sac membrane for 3 eggs at two stages (E3 and E4) to characterize the spatial distribution of the complexity of the vascular network.

RESULTS

With the proposed method, the spatial distribution of the complexity of the yolk sac membrane vascular network was visualized. From E3 to E4, the local fractal dimension increased in 3 eggs, 1.80 ± 0.02 vs. 1.85 ± 0.02, 1.72 ± 0.03 vs. 1.83 ± 0.02, and 1.77 ± 0.03 vs. 1.82 ± 0.02, respectively. The mean local fractal dimension in the most distal area from the embryo proper was the lowest at E3 while the highest at E4. At E3, the most peaks of the local fractal dimension were located in the vein territories and shifted to artery territories at E4.

CONCLUSIONS

The spatial distribution of the complexity of the yolk sac membrane vascular network exhibited diverse patterns at different stages. In addition from E3 to E4, the increment of complexity at the intersection areas between arteries and sinus terminalis was with the most advance. This is consistent with the physiologic evidence. The present work provides a potential approach for investigating the spatiotemporal pattern of the angiogenic process.

Association Between Retinal Microvascular Metrics Using Optical Coherence Tomography Angiography and Carotid Artery Stenosis in a Chinese Cohort

Objectives: The main aim was to investigate the association between retinal microvascular metrics using optical coherence tomography angiography (OCTA) and carotid artery stenosis (CAS) in an aging Chinese cohort.

Methods: In this cross-sectional and observational study, 138 eyes of 138 participants were examined. Indices of the microcirculation measured by OCTA included mean vessel density (VD), skeleton density (SD), vessel diameter index (VDI), fractal dimension (FD) and foveal avascular zone (FAZ) of the superficial retinal layer (SRL) and deep retinal layer (DRL), and peripapillary vessel caliber. The correlation of these indices with the carotid atherosclerotic lesions including carotid intima media thickness (CIMT) and common carotid artery (CCA) plaque was assessed.

Results: A total of 72 of 138 eyes demonstrated an increased (≥1 mm) CIMT, and 32 of the eyes presented common carotid plaques. Macular VD, SD, and FD were decreased with the increasing CCA caliber diameter (p < 0.05, respectively). Superficial and deep macular FDs were negatively associated with CIMT as well as the existence of CCA plaques (p < 0.05, respectively).

Conclusion: Changes in retinal microvasculature accessed by OCTA may be used as one of the non-invasive early indicators to monitor asymptomatic CAS.

Estimating the Fractal Dimensions of Vascular Networks and Other Branching Structures: Some Words of Caution

Branching patterns are ubiquitous in nature; consequently, over the years many researchers have tried to characterize the complexity of their structures. Due to their hierarchical nature and resemblance to fractal trees, they are often thought to have fractal properties; however, their non-homogeneity (i.e., lack of strict self-similarity) is often ignored. In this paper we review and examine the use of the box-counting and sandbox methods to estimate the fractal dimensions of branching structures. We highlight the fact that these methods rely on an assumption of self-similarity that is not present in branching structures due to their non-homogeneous nature. Looking at the local slopes of the log–log plots used by these methods reveals the problems caused by the non-homogeneity. Finally, we examine the role of the canopies (endpoints or limit points) of branching structures in the estimation of their fractal dimensions.

Development of Novel Fractal Method for Characterizing the Distribution of Blood Flow in Multi-Scale Vascular Tree

Blood perfusion is an important index for the function of the cardiovascular system and it can be indicated by the blood flow distribution in the vascular tree. As the blood flow in a vascular tree varies in a large range of scales and fractal analysis owns the ability to describe multi-scale properties, it is reasonable to apply fractal analysis to depict the blood flow distribution. The objective of this study is to establish fractal methods for analyzing the blood flow distribution which can be applied to real vascular trees. For this purpose, the modified methods in fractal geometry were applied and a special strategy was raised to make sure that these methods are applicable to an arbitrary vascular tree. The validation of the proposed methods on real arterial trees verified the ability of the produced parameters (fractal dimension and multifractal spectrum) in distinguishing the blood flow distribution under different physiological states. Furthermore, the physiological significance of the fractal parameters was investigated in two situations. For the first situation, the vascular tree was set as a perfect binary tree and the blood flow distribution was adjusted by the split ratio. As the split ratio of the vascular tree decreases, the fractal dimension decreases and the multifractal spectrum expands. The results indicate that both fractal parameters can quantify the degree of blood flow heterogeneity. While for the second situation, artificial vascular trees with different structures were constructed and the hemodynamics in these vascular trees was simulated. The results suggest that both the vascular structure and the blood flow distribution affect the fractal parameters for blood flow. The fractal dimension declares the integrated information about the heterogeneity of vascular structure and blood flow distribution. In contrast, the multifractal spectrum identifies the heterogeneity features in blood flow distribution or vascular structure by its width and height. The results verified that the proposed methods are capable of depicting the multi-scale features of the blood flow distribution in the vascular tree and further are potential for investigating vascular physiology.

Morphological characterization of Etv2 vascular explants using fractal analysis and atomic force microscopy

The rapid engraftment of vascular networks is critical for functional incorporation of tissue explants. However, existing methods for inducing angiogenesis utilize approaches that yield vasculature with poor temporal stability or inadequate mechanical integrity, which reduce their robustness in vivo. The transcription factor Ets variant 2 (Etv2) specifies embryonic hematopoietic and vascular endothelial cell (EC) development, and is transiently reactivated during postnatal vascular regeneration and tumor angiogenesis. This study investigates the role for Etv2 upregulation in forming stable vascular beds both in vitro and in vivo. Control and Etv2⁺ prototypical fetal-derived human umbilical vein ECs (HUVECs) and adult ECs were angiogenically grown into vascular beds. These vessel beds were characterized using fractal dimension and lacunarity, to quantify their branching complexity and space-filling homogeneity, respectively. Atomic force microscopy (AFM) was used to explore whether greater complexity and homogeneity lead to more mechanically stable vessels. Additionally, markers of EC integrity were used to probe for mechanistic clues. Etv2⁺ HUVECs exhibit greater branching, vessel density, and structural homogeneity, and decreased stiffness in vitro and in vivo, indicating a greater propensity for stable vessel formation. When co-cultured with colon tumor organoid tissue, Etv2⁺ HUVECs had decreased fractal dimension and lacunarity compared to Etv2⁺ HUVECs cultured alone, indicating that vessel density and homogeneity of vessel spacing increased due to the presence of Etv2. This study sets forth the novel concept that fractal dimension, lacunarity, and AFM are as informative as conventional angiogenic measurements, including vessel branching and density, to assess vascular perfusion and stability.

Fractal dimension of diffusion-limited aggregation clusters grown on spherical surfaces

In this work we study the fractal properties of diffusion-limited aggregation (DLA) clusters grown on spherical surfaces. Diffusion-limited aggregation clusters, or DLA trees, are highly branched fractal clusters formed by the adhesion of particles. In two-dimensional media, DLA clusters have a fractal dimension D_{f}=1.70 in the continuous limit. In some physical systems, the existence of characteristic lengths leads us to model them as discrete systems. Such characteristic lengths may result also from limitations in measuring instruments, for example, the resolution of biomedical imaging systems. We simulate clusters for different particle sizes and examine the influence of discretization by exploring the systems in terms of the relationship between the particle size r and the radius of the sphere R. We also study the effect of stereographic projection on the fractal properties of DLA clusters. Both discretization and projection alter the fractal dimension of DLA clusters grown on curved surfaces and must be considered in the interpretation of photographic biomedical images.