Fractal analysis of vascular networks: Insights from morphogenesis

In Silico Design of Heterogeneous Microvascular Trees Using Generative Adversarial Networks and Constrained Constructive Optimization

OBJECTIVE

Designing physiologically adequate microvascular trees is of crucial relevance for bioengineering functional tissues and organs. Yet, currently available methods are poorly suited to replicate the morphological and topological heterogeneity of real microvascular trees because the parameters used to control tree generation are too simplistic to mimic results of the complex angiogenetic and structural adaptation processes in vivo.

METHODS

We propose a method to overcome this limitation by integrating a conditional deep convolutional generative adversarial network (cDCGAN) with a local fractal dimension-oriented constrained constructive optimization (LFDO-CCO) strategy. The cDCGAN learns the patterns of real microvascular bifurcations allowing for their artificial replication. The LFDO-CCO strategy connects the generated bifurcations hierarchically to form microvascular trees with a vessel density corresponding to that observed in healthy tissues.

RESULTS

The generated artificial microvascular trees are consistent with real microvascular trees regarding characteristics such as fractal dimension, vascular density, and coefficient of variation of diameter, length, and tortuosity.

CONCLUSIONS

These results support the adoption of the proposed strategy for the generation of artificial microvascular trees in tissue engineering as well as for computational modeling and simulations of microcirculatory physiology.

Modeling oxygen transport in the brain: An efficient coarse-grid approach to capture perivascular gradients in the parenchyma

Recent progresses in intravital imaging have enabled highly-resolved measurements of periarteriolar oxygen gradients (POGs) within the brain parenchyma. POGs are increasingly used as proxies to estimate the local baseline oxygen consumption, which is a hallmark of cell activity. However, the oxygen profile around a given arteriole arises from an interplay between oxygen consumption and delivery, not only by this arteriole but also by distant capillaries. Integrating such interactions across scales while accounting for the complex architecture of the microvascular network remains a challenge from a modelling perspective. This limits our ability to interpret the experimental oxygen maps and constitutes a key bottleneck toward the inverse determination of metabolic rates of oxygen. We revisit the problem of parenchymal oxygen transport and metabolism and introduce a simple, conservative, accurate and scalable direct numerical method going beyond canonical Krogh-type models and their associated geometrical simplifications. We focus on a two-dimensional formulation, and introduce the concepts needed to combine an operator-splitting and a Green's function approach. Oxygen concentration is decomposed into a slowly-varying contribution, discretized by Finite Volumes over a coarse cartesian grid, and a rapidly-varying contribution, approximated analytically in grid-cells surrounding each vessel. Starting with simple test cases, we thoroughly analyze the resulting errors by comparison with highly-resolved simulations of the original transport problem, showing considerable improvement of the computational-cost/accuracy balance compared to previous work. We then demonstrate the model ability to flexibly generate synthetic data reproducing the spatial dynamics of oxygen in the brain parenchyma, with sub-grid resolution. Based on these synthetic data, we show that capillaries distant from the arteriole cannot be overlooked when interpreting POGs, thus reconciling recent measurements of POGs across cortical layers with the fundamental idea that variations of vascular density within the depth of the cortex may reveal underlying differences in neuronal organization and metabolic load.

Orthotopic transplantation of the bioengineered lung using a mouse-scale perfusion-based bioreactor and human primary endothelial cells

Whole lung engineering and the transplantation of its products is an ambitious goal and ultimately a viable solution for alleviating the donor-shortage crisis for lung transplants. There are several limitations currently impeding progress in the field with a major obstacle being efficient revascularization of decellularized scaffolds, which requires an extremely large number of cells when using larger pre-clinical animal models. Here, we developed a simple but effective experimental pulmonary bioengineering platform by utilizing the lung as a scaffold. Revascularization of pulmonary vasculature using human umbilical cord vein endothelial cells was feasible using a novel in-house developed perfusion-based bioreactor. The endothelial lumens formed in the peripheral alveolar area were confirmed using a transmission electron microscope. The quality of engineered lung vasculature was evaluated using box-counting analysis of histological images. The engineered mouse lungs were successfully transplanted into the orthotopic thoracic cavity. The engineered vasculature in the lung scaffold showed blood perfusion after transplantation without significant hemorrhage. The mouse-based lung bioengineering system can be utilized as an efficient ex-vivo screening platform for lung tissue engineering.

Enhanced perfusion following exposure to radiotherapy: A theoretical investigation

Tumour angiogenesis leads to the formation of blood vessels that are structurally and spatially heterogeneous. Poor blood perfusion, in conjunction with increased hypoxia and oxygen heterogeneity, impairs a tumour's response to radiotherapy. The optimal strategy for enhancing tumour perfusion remains unclear, preventing its regular deployment in combination therapies. In this work, we first identify vascular architectural features that correlate with enhanced perfusion following radiotherapy, using in vivo imaging data from vascular tumours. Then, we present a novel computational model to determine the relationship between these architectural features and blood perfusion in silico. If perfusion is defined to be the proportion of vessels that support blood flow, we find that vascular networks with small mean diameters and large numbers of angiogenic sprouts show the largest increases in perfusion post-irradiation for both biological and synthetic tumours. We also identify cases where perfusion increases due to the pruning of hypoperfused vessels, rather than blood being rerouted. These results indicate the importance of considering network composition when determining the optimal irradiation strategy. In the future, we aim to use our findings to identify tumours that are good candidates for perfusion enhancement and to improve the efficacy of combination therapies.

A Self-Similarity Logic May Shape the Organization of the Nervous System

From the morphological point of view, the nervous system exhibits a fractal, self-similar geometry at various levels of observations, from single cells up to cell networks. From the functional point of view, it is characterized by a hierarchical organization in which self-similar structures (networks) of different miniaturizations are nested within each other. In particular, neuronal networks, interconnected to form neuronal systems, are formed by neurons, which operate thanks to their molecular networks, mainly having proteins as components that via protein-protein interactions can be assembled in multimeric complexes working as micro-devices. On this basis, the term "self-similarity logic" was introduced to describe a nested organization where, at the various levels, almost the same rules (logic) to perform operations are used. Self-similarity and self-similarity logic both appear to be intimately linked to the biophysical evidence for the nervous system being a pattern-forming system that can flexibly switch from one coherent state to another. Thus, they can represent the key concepts to describe its complexity and its concerted, holistic behavior.

Fractal Analysis in Clinical Neurosciences: An Overview

Over the last years, fractals have entered into the realms of clinical neurosciences. The whole brain and its components (i.e., neurons and astrocytes) have been studied as fractal objects, and even more relevant, the fractal-based quantification of the geometrical complexity of histopathological and neuroradiological images as well as neurophysiopathological time series has suggested the existence of a gradient in the pattern representation of neurological diseases. Computational fractal-based parameters have been suggested as potential diagnostic and prognostic biomarkers in different brain diseases, including brain tumors, neurodegeneration, epilepsy, demyelinating diseases, cerebrovascular malformations, and psychiatric disorders as well. This chapter and the entire third section of this book are focused on practical applications of computational fractal-based analysis into the clinical neurosciences, namely, neurology and neuropsychiatry, neuroradiology and neurosurgery, neuropathology, neuro-oncology and neurorehabilitation, neuro-ophthalmology, and cognitive neurosciences, with special emphasis on the translation of the fractal dimension and other fractal parameters as clinical biomarkers useful from bench to bedside.

Fractal-Based Analysis of Histological Features of Brain Tumors

The structural complexity of brain tumor tissue represents a major challenge for effective histopathological diagnosis. Tumor vasculature is known to be heterogeneous, and mixtures of patterns are usually present. Therefore, extracting key descriptive features for accurate quantification is not a straightforward task. Several steps are involved in the texture analysis process where tissue heterogeneity contributes to the variability of the results. One of the interesting aspects of the brain lies in its fractal nature. Many regions within the brain tissue yield similar statistical properties at different scales of magnification. Fractal-based analysis of the histological features of brain tumors can reveal the underlying complexity of tissue structure and angiostructure, also providing an indication of tissue abnormality development. It can further be used to quantify the chaotic signature of disease to distinguish between different temporal tumor stages and histopathological grades.Brain meningioma subtype classifications' improvement from histopathological images is the main focus of this chapter. Meningioma tissue texture exhibits a wide range of histological patterns whereby a single slide may show a combination of multiple patterns. Distinctive fractal patterns quantified in a multiresolution manner would be for better spatial relationship representation. Fractal features extracted from textural tissue patterns can be useful in characterizing meningioma tumors in terms of subtype classification, a challenging problem compared to histological grading, and furthermore can provide an objective measure for quantifying subtle features within subtypes that are hard to discriminate.

Computational Fractal-Based Analysis of Brain Tumor Microvascular Networks

Brain parenchyma microvasculature is set in disarray in the presence of tumors, and malignant brain tumors are among the most vascularized neoplasms in humans. As microvessels can be easily identified in histologic specimens, quantification of microvascularity can be used alone or in combination with other histological features to increase the understanding of the dynamic behavior, diagnosis, and prognosis of brain tumors. Different brain tumors, and even subtypes of the same tumor, show specific microvascular patterns, as a kind of "microvascular fingerprint," which is particular to each histotype. Reliable morphometric parameters are required for the qualitative and quantitative characterization of the neoplastic angioarchitecture, although the lack of standardization of a technique able to quantify the microvascular patterns in an objective way has limited the "morphometric approach" in neuro-oncology.In this chapter, we focus on the importance of computational-based morphometrics, for the objective description of tumoral microvascular fingerprinting. By also introducing the concept of "angio-space," which is the tumoral space occupied by the microvessels, we here present fractal analysis as the most reliable computational tool able to offer objective parameters for the description of the microvascular networks.The spectrum of different angioarchitectural configurations can be quantified by means of Euclidean and fractal-based parameters in a multiparametric analysis, aimed to offer surrogate biomarkers of cancer. Such parameters are here described from the methodological point of view (i.e., feature extraction) as well as from the clinical perspective (i.e., relation to underlying physiology), in order to offer new computational parameters to the clinicians with the final goal of improving diagnostic and prognostic power of patients affected by brain tumors.

A spatially distributed model of brain metabolism highlights the role of diffusion in brain energy metabolism

The different active roles of neurons and astrocytes during neuronal activation are associated with the metabolic processes necessary to supply the energy needed for their respective tasks at rest and during neuronal activation. Metabolism, in turn, relies on the delivery of metabolites and removal of toxic byproducts through diffusion processes and the cerebral blood flow. A comprehensive mathematical model of brain metabolism should account not only for the biochemical processes and the interaction of neurons and astrocytes, but also the diffusion of metabolites. In the present article, we present a computational methodology based on a multidomain model of the brain tissue and a homogenization argument for the diffusion processes. In our spatially distributed compartment model, communication between compartments occur both through local transport fluxes, as is the case within local astrocyte-neuron complexes, and through diffusion of some substances in some of the compartments. The model assumes that diffusion takes place in the extracellular space (ECS) and in the astrocyte compartment. In the astrocyte compartment, the diffusion across the syncytium network is implemented as a function of gap junction strength. The diffusion process is implemented numerically by means of a finite element method (FEM) based spatial discretization, and robust stiff solvers are used to time integrate the resulting large system. Computed experiments show the effects of ECS tortuosity, gap junction strength and spatial anisotropy in the astrocyte network on the brain energy metabolism.

Modeling the response of homogeneous and heterogeneous cerebral capillary networks to local changes in vessel diameters

While microvascular cerebral capillary networks are known to be highly heterogeneous, previous computational models have predicted that heterogeneous cerebral capillary flow patterns result in lower brain tissue partial oxygen pressures. Moreover, as blood flow increases, the flux among capillaries homogenizes. This homogenization of flow is expected to improve the efficiency of oxygenation extraction from the blood. In this work, we use mathematical modeling to explore a possible functional role for the high degree of heterogeneity observed in cerebral capillary networks. Our results suggest that heterogeneity allows for a greater response of tissue oxygen levels to local changes in vessel diameters due to neuronal activation. This result is confirmed for a full 3-dimensional model of capillary networks that includes oxygen diffusion within the tissue region and a reduced model that accounts for changes in capillary blood flow.

Computational models for generating microvascular structures: Investigations beyond medical imaging resolution

Angiogenesis, arteriogenesis, and pruning are revascularization processes essential to our natural vascular development and adaptation, as well as central players in the onset and development of pathologies such as tumoral growth and stroke recovery. Computational modeling allows for repeatable experimentation and exploration of these complex biological processes. In this review, we provide an introduction to the biological understanding of the vascular adaptation processes of sprouting angiogenesis, intussusceptive angiogenesis, anastomosis, pruning, and arteriogenesis, discussing some of the more significant contributions made to the computational modeling of these processes. Each computational model represents a theoretical framework for how biology functions, and with rises in computing power and study of the problem these frameworks become more accurate and complete. We highlight physiological, pathological, and technological applications that can be benefit from the advances performed by these models, and we also identify which elements of the biology are underexplored in the current state-of-the-art computational models. This article is categorized under: Cancer > Computational Models Cardiovascular Diseases > Computational Models.

A Computational Template for Three-Dimensional Modeling of the Vascular Scaffold of the Human Thyroid Gland

We recently designed an innovative scaffold-bioreactor unit for the bioengineering of a three-dimensional (3D) bioartificial human thyroid gland or its miniaturized replica as a part of a microfluidic chip test system. This device is based on the evidence that the 3D geometry of the intraglandular stromal/vascular scaffold (SVS; i.e., the fibrous and vascular matrix) of mammalian viscera plays a key role in guiding growth and differentiation of in vitro seeded cells. Therefore, we initiated a research program focused on computer-aided reconstruction of the 2nd to 4th order intralobar arterial network (IAN) of the human thyroid gland as a reliable surrogate for its 3D SVS, to be used as an input for rapid prototyping of a biomaterial replica. To this end, we developed a computational template that works within the Mathematica environment, giving rise to a quasi-fractal growth of the IAN distribution, constrained within an approximation of the thyroid lobe shape as a closed surface. Starting from edge detection of planar images of real human thyroid lobes acquired by in vivo real-time ultrasonography, we performed data approximation of the lobar profiles based on splines and Bezier curves, providing 3D lobar shapes as geometric boundaries for vessel growth by a diffusion-limited aggregation model. Our numerical procedures allowed for a robust connection between development of lobar arterial trees and thyroid lobe shape, led to a vascular self-similarity consistent with that of a cadaveric lobar arterial cast, and reproduced arterial vessels in a proportion not statistically different from that described for the real human thyroid gland. We conclude that our algorithmic template offers a reliable reproduction of the extremely complex IAN of the adult human thyroid lobe, potentially useful as a computational guidance for bioprinting of thyroid lobe matrix replicas. In addition, due to the simplicity and limited number of morphometrical parameters required by our system, we predict its application to the design of a number of patient-tailored human bioartificial organs and organs-on-chip, including parenchymal viscera and bones. Impact statement The study introduces the computer simulation of the three-dimensional (3D) intrinsic vascular matrix of the human thyroid gland, offering a general concept applicable to a number of other human viscera. Indeed, it provides a flexible software tool for reproduction of a 3D surrogate of the organ's 3D stromal matrix, suitable for eventual 3D bioprinting with biomaterials, and recellularization with organ-specific stem cells/progenitors. The final expectation is the design of patient-tailored 3D organ's matrices upon clinical request.

Topological morphogenesis of neuroepithelial organoids

Animal organs exhibit complex topologies involving cavities and tubular networks, which underlie their form and function^1-3. However, how topology emerges during the development of organ shape, or morphogenesis, remains elusive. Here we combine tissue reconstitution and quantitative microscopy to show that tissue topology and shape is governed by two distinct modes of topological transitions^4,5. One mode involves the fusion of two separate epithelia and the other involves the fusion of two ends of the same epithelium. The morphological space is captured by a single control parameter that can be traced back to the relative rates of the two epithelial fusion modes. Finally, we identify a pharmacologically accessible pathway that regulates the frequency of two modes of epithelial fusion, and demonstrate the control of organoid topology and shape. The physical principles uncovered here provide fundamental insights into the self-organization of complex tissues⁶.

Diploic vein morphology in normal and craniosynostotic adult human skulls

Diploic veins (DV) run within the cranial diploe, where they leave channels that can be studied in osteological samples. This study investigates overall DV variability in human adults and the effects of sex, age, cranial dimensions, and dysmorphogenesis associated with craniosynostosis (CS). The morphology of macroscopic diploic channels was analyzed in a set of the qualitative and quantitative variables in computed tomography-images of crania of anatomically normal and craniosynostotic adult individuals. Macroscopic diploic channels occur most frequently in the frontal and parietal bones, often with a bilaterally symmetrical pattern. DV-features (especially DV-pattern) are characterized by high individual diversity. On average, there are 5.4 ± 3.5 large macroscopic channels (with diameters >1 mm) per individual, with a mean diameter of 1.7 ± 0.4 mm. Age and sex have minor effects on DV, and cranial proportions significantly influence DV only in CS skulls. CS is associated with changes in the DV numbers, distributions, and diameters. Craniosynostotic skulls, especially brachycephalic skulls, generally present smaller DV diameters, and dolichocephalic skulls display increased number of frontal DV. CS, associated with altered cranial dimensions, suture imbalance, increased intracranial pressure, and with changes of the endocranial craniovascular system, significantly also affects the macroscopic morphology of DV in adults, in terms of both structural (topological redistribution) and functional factors. The research on craniovascular morphology and CS may be of interest in biological anthropology, paleopathology, medicine (e.g., surgical planning), but also in zoology and paleontology.

Tension-Relaxation In Vivo Computing Principle for Tumor Sensitization and Targeting

By modeling the tumor sensitization and targeting (TST) as a natural computational process, we have proposed the framework of nanorobots-assisted in vivo computation. The externally manipulable nanorobots are steered to detect the tumor in the high-risk tissue, which is analogous to the process of searching for the optimal solution by the computing agents in the search space. To overcome the constraint of a nanorobotic platform that can only generate a uniform magnetic field to actuate the nanorobots, we have proposed the weak priority evolution strategy (WP-ES) in our previous works. However, these works do not consider the proportions of the nanorobot control and tracking operations, which are part and parcel of in vivo computation as the control operation aims at searching for the tumor effectively while the tracking mode is used for gathering information about the biological gradient function (BGF). Careful planning about the durations spent in these operations is needed for optimal performance of the TST strategy. To account for this issue, in the current article, we propose a novel computational principle, called the tension-relaxation (T-R) principle, to balance the displacements of nanorobots during each control and tracking cycle. Furthermore, we build three tumor vascular models with different sizes to represent three different targeting regions as the morphology of tumor vasculature determined by the tumor growth process is an important factor affecting TST. We carry out the computational experiments for tumors with three different sizes for several representative landscapes by introducing the T-R principle into the WP-ES-based swarm intelligence algorithms and considering the realistic internal constraints. The experimental outcomes demonstrate the effectiveness of the proposed TST strategy.

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.

Network-driven anomalous transport is a fundamental component of brain microvascular dysfunction

Blood microcirculation supplies neurons with oxygen and nutrients, and contributes to clearing their neurotoxic waste, through a dense capillary network connected to larger tree-like vessels. This complex microvascular architecture results in highly heterogeneous blood flow and travel time distributions, whose origin and consequences on brain pathophysiology are poorly understood. Here, we analyze highly-resolved intracortical blood flow and transport simulations to establish the physical laws governing the macroscopic transport properties in the brain micro-circulation. We show that network-driven anomalous transport leads to the emergence of critical regions, whether hypoxic or with high concentrations of amyloid-β, a waste product centrally involved in Alzheimer's Disease. We develop a Continuous-Time Random Walk theory capturing these dynamics and predicting that such critical regions appear much earlier than anticipated by current empirical models under mild hypoperfusion. These findings provide a framework for understanding and modelling the impact of microvascular dysfunction in brain diseases, including Alzheimer's Disease.

Parallel generation of extensive vascular networks with application to an archetypal human kidney model

Given the relevance of the inextricable coupling between microcirculation and physiology, and the relation to organ function and disease progression, the construction of synthetic vascular networks for mathematical modelling and computer simulation is becoming an increasingly broad field of research. Building vascular networks that mimic in vivo morphometry is feasible through algorithms such as constrained constructive optimization (CCO) and variations. Nevertheless, these methods are limited by the maximum number of vessels to be generated due to the whole network update required at each vessel addition. In this work, we propose a CCO-based approach endowed with a domain decomposition strategy to concurrently create vascular networks. The performance of this approach is evaluated by analysing the agreement with the sequentially generated networks and studying the scalability when building vascular networks up to 200 000 vascular segments. Finally, we apply our method to vascularize a highly complex geometry corresponding to the cortex of a prototypical human kidney. The technique presented in this work enables the automatic generation of extensive vascular networks, removing the limitation from previous works. Thus, we can extend vascular networks (e.g. obtained from medical images) to pre-arteriolar level, yielding patient-specific whole-organ vascular models with an unprecedented level of detail.

Digital Reconstruction of the Neuro-Glia-Vascular Architecture

Astrocytes connect the vasculature to neurons mediating the supply of nutrients and biochemicals. They are involved in a growing number of physiological and pathophysiological processes that result from biophysical, physiological, and molecular interactions in this neuro-glia-vascular ensemble (NGV). The lack of a detailed cytoarchitecture severely restricts the understanding of how they support brain function. To address this problem, we used data from multiple sources to create a data-driven digital reconstruction of the NGV at micrometer anatomical resolution. We reconstructed 0.2 mm3 of the rat somatosensory cortex with 16 000 morphologically detailed neurons, 2500 protoplasmic astrocytes, and its microvasculature. The consistency of the reconstruction with a wide array of experimental measurements allows novel predictions of the NGV organization, allowing the anatomical reconstruction of overlapping astrocytic microdomains and the quantification of endfeet connecting each astrocyte to the vasculature, as well as the extent to which they cover the latter. Structural analysis showed that astrocytes optimize their positions to provide uniform vascular coverage for trophic support and signaling. However, this optimal organization rapidly declines as their density increases. The NGV digital reconstruction is a resource that will enable a better understanding of the anatomical principles and geometric constraints, which govern how astrocytes support brain function.

Imaging the Retinal Vasculature

Advances in retinal imaging are enabling researchers and clinicians to make precise noninvasive measurements of the retinal vasculature in vivo. This includes measurements of capillary blood flow, the regulation of blood flow, and the delivery of oxygen, as well as mapping of perfused blood vessels. These advances promise to revolutionize our understanding of vascular regulation, as well as the management of retinal vascular diseases. This review provides an overview of imaging and optical measurements of the function and structure of the ocular vasculature. We include general characteristics of vascular systems with an emphasis on the eye and its unique status. The functions of vascular systems are discussed, along with physical principles governing flow and its regulation. Vascular measurement techniques based on reflectance and absorption are briefly introduced, emphasizing ways of generating contrast. One of the prime ways to enhance contrast within vessels is to use techniques sensitive to the motion of cells, allowing precise measurements of perfusion and blood velocity. Finally, we provide a brief introduction to retinal vascular diseases.

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.

Assessment of Vascular Network Connectivity of Hepatocellular Carcinoma Using Graph-Based Approach

BACKGROUND

The angiogenesis of liver cancer is a key condition for its growth, invasion, and metastasis. This study aims to investigate vascular network connectivity of hepatocellular carcinoma (HCC) using graph-based approach.

METHODS

Orthotopic HCC xenograft models (n=10) and the healthy controls (n=10) were established. After 21 days of modeling, hepatic vascular casting and Micro-CT scanning were performed for angiography, followed by blood vessels automatic segmentation and vascular network modeling. The topologic parameters of vascular network, including clustering coefficient (CC), network structure entropy (NSE), and average path length (APL) were quantified. Topologic parameters of the tumor region, as well as the background liver were compared between HCC group and normal control group.

RESULTS

Compared with normal control group, the tumor region of HCC group showed significantly decreased CC [(0.046 ± 0.005) vs. (0.052 ± 0.006), P=0.026], and NSE [(0.9894 ± 0.0015) vs. (0.9927 ± 0.0010), P<0.001], and increased APL [(0.433 ± 0.138) vs. (0.188 ± 0.049), P<0.001]. Compared with normal control group, the background liver of HCC group showed significantly decreased CC [(0.047 ± 0.004) vs. (0.052 ± 0.006), P=0.041] and increased NSE [0.9938 (0.9936~0.9940) vs. (0.9927 ± 0.0010), P=0.035]. No significant difference was identified for APL between the two groups.

CONCLUSION

Graph-based approach allows quantification of vascular connectivity of HCC. Disrupted vascular topological connectivity exists in the tumor region, as well as the background liver of HCC.

The severity of microstrokes depends on local vascular topology and baseline perfusion

Cortical microinfarcts are linked to pathologies like cerebral amyloid angiopathy and dementia. Despite their relevance for disease progression, microinfarcts often remain undetected and the smallest scale of blood flow disturbance has not yet been identified. We employed blood flow simulations in realistic microvascular networks from the mouse cortex to quantify the impact of single-capillary occlusions. Our simulations reveal that the severity of a microstroke is strongly affected by the local vascular topology and the baseline flow rate in the occluded capillary. The largest changes in perfusion are observed in capillaries with two inflows and two outflows. This specific topological configuration only occurs with a frequency of 8%. The majority of capillaries have one inflow and one outflow and is likely designed to efficiently supply oxygen and nutrients. Taken together, microstrokes bear potential to induce a cascade of local disturbances in the surrounding tissue, which might accumulate and impair energy supply locally.

Mathematical modeling of intraplaque neovascularization and hemorrhage in a carotid atherosclerotic plaque

BACKGROUND

Growing experimental evidence has identified neovascularization from the adventitial vasa vasorum and induced intraplaque hemorrhage (IPH) as critical indicators during the development of vulnerable atherosclerotic plaques. In this study, we propose a mathematical model incorporating intraplaque angiogenesis and hemodynamic calculation of the microcirculation, to obtain the quantitative evaluation of the influences of intraplaque neovascularization and hemorrhage on vulnerable plaque development. A two-dimensional nine-point model of angiogenic microvasculature is generated based on the histology of a patient's carotid plaque. The intraplaque angiogenesis model includes three key cells (endothelial cells, smooth muscle cells, and macrophages) and three key chemical factors (vascular endothelial growth factors, extracellular matrix, and matrix metalloproteinase), which densities and concentrations are described by a series of reaction-diffusion equations. The hemodynamic calculation by coupling the intravascular blood flow, the extravascular plasma flow, and the transvascular transport is carried out on the generated angiogenic microvessel network. We then define the IPH area by using the plasma concentration in the interstitial tissue, as well as the extravascular transport across the capillary wall.

RESULTS

The simulational results reproduce a series of pathophysiological phenomena during the atherosclerotic plaque progression. It is found that the high microvessel density region at the shoulder areas and the extravascular flow across the leaky wall of the neovasculature contribute to the IPH observed widely in vulnerable plaques. The simulational results are validated by both the in vivo MR imaging data and in vitro experimental observations and show significant consistency in quantity ground. Moreover, the sensitivity analysis of model parameters reveals that the IPH area and extent can be reduced significantly by decreasing the MVD and the wall permeability of the neovasculature.

CONCLUSIONS

The current quantitative model could help us to better understand the roles of microvascular and intraplaque hemorrhage during the carotid plaque progression.

The capillary fascicle in skeletal muscle: Structural and functional physiology of RBC distribution in capillary networks

KEY POINTS

The capillary module, consisting of parallel capillaries from arteriole to venule, is classically considered as the building block of complex capillary networks. In skeletal muscle, this structure fails to address how blood flow is regulated along the entire length of the synchronously contracting muscle fibres. Using intravital video microscopy of resting extensor digitorum longus muscle in rats, we demonstrated the capillary fascicle as a series of interconnected modules forming continuous columns that align naturally with the dimensions of the muscle fascicle. We observed structural heterogeneity for module topology, and functional heterogeneity in space and time for capillary-red blood cell (RBC) haemodynamics within a module and between modules. We found that module RBC haemodynamics were independent of module resistance, providing direct evidence for microvascular flow regulation at the level of the capillary module. The capillary fascicle is an updated paradigm for characterizing blood flow and RBC distribution in skeletal muscle capillary networks.

ABSTRACT

Capillary networks are the fundamental site of oxygen exchange in the microcirculation. The capillary module (CM), consisting of parallel capillaries from terminal arteriole (TA) to post-capillary venule (PCV), is classically considered as the building block of complex capillary networks. In skeletal muscle, this structure fails to address how blood flow is regulated along the entire length of the synchronously contracting muscle fibres, requiring co-ordination from numerous modules. It has previously been recognized that TAs and PCVs interact with multiple CMs, creating interconnected networks. Using label-free intravital video microscopy of resting extensor digitorum longus muscle in rats, we found that these networks form continuous columns of linked CMs spanning thousands of microns, herein denoted as the capillary fascicle (CF); this structure aligns naturally with the dimensions of the muscle fascicle. We measured capillary-red blood cell (RBC) haemodynamics and module topology (n = 9 networks, 327 modules, 1491 capillary segments). The average module had length 481 μm, width 157 μm and 9.51 parallel capillaries. We observed structural heterogeneity for CM topology, and functional heterogeneity in space and time for capillary-RBC haemodynamics within a module and between modules. There was no correlation between capillary RBC velocity and lineal density. A passive inverse relationship between module length and haemodynamics was remarkably absent, providing direct evidence for microvascular flow regulation at the level of the CM. In summary, the CF is an updated paradigm for characterizing RBC distribution in skeletal muscle, and strengthens the theory of capillary networks as major contributors to the signal that regulates capillary perfusion.

Wavelength of a Turing-type mechanism regulates the morphogenesis of meshwork patterns

The meshwork pattern is a significant pattern in the development of biological tissues and organs. It is necessary to explore the mathematical mechanism of meshwork pattern formation. In this paper, we found that the meshwork pattern is formed by four kinds of stalk behaviours: stalk extension, tip bifurcation, side branching and tip fusion. The Turing-type pattern underlying the meshwork pattern is a Turing spot pattern, which indicates that the Turing instability of the spot pattern promotes activator peak formation and then guides the formation of meshwork patterns. Then, we found that the Turing wavelength decreased in turn from tip bifurcation to side branching to tip fusion via statistical evaluation. Through the functional relationship between the Turing wavelength and model parameters ([Formula: see text] and [Formula: see text]), we found that parameters [Formula: see text] and [Formula: see text] had monotonic effects on the Turing wavelength and that parameter [Formula: see text] had nonmonotonic effects. Furthermore, we performed simulations of local meshwork pattern formation under variable model parameter values. The simulation results verified the corresponding relationship between the Turing wavelength and stalk behaviours and the functional relationship between the Turing wavelength and model parameters. The simulation results showed that the Turing wavelength regulated the meshwork pattern and that the small Turing wavelength facilitated dense meshwork pattern formation. Our work provides novel insight into and understanding of the formation of meshwork patterns. We believe that studies associated with network morphogenesis can benefit from our work.

Remodeling mechanisms determine size distributions in developing retinal vasculature

The development of retinal blood vessels has extensively been used as a model to study vascular pattern formation. To date, various quantitative measurements, such as size distribution have been performed, but the relationship between pattern formation mechanisms and these measurements remains unclear. In the present study, we first focus on the islands (small regions subdivided by the capillary network). We quantitatively measured the island size distribution in the retinal vascular network and found that it tended to exhibit an exponential distribution. We were able to recapitulate this distribution pattern in a theoretical model by implementing the stochastic disappearance of vessel segments around arteries could reproduce the observed exponential distribution of islands. Second, we observed that the diameter distribution of the retinal artery segment obeyed a power law. We theoretically showed that an equal bifurcation branch pattern and Murray’s law could reproduce this pattern. This study demonstrates the utility of examining size distribution for understanding the mechanisms of vascular pattern formation.

The Fractal and GLCM Textural Parameters of Chromatin May Be Potential Biomarkers of Papillary Thyroid Carcinoma in Hashimoto's Thyroiditis Specimens

Occasionally, Hashimoto's thyroiditis (HT) and papillary thyroid carcinoma (PTC) share similar nuclear features. The current study aims to quantify the differences between the investigated specimens of HT-associated PTC versus the HT alone, to reduce the subjective experience of an observer, by the use of fractal parameters as well as gray-level co-occurrence matrix (GLCM) textural parameters. We have analyzed 250 segmented nuclei per group (nn = 25 per patient and np = 10 patients per group) using the ImageJ software (NIH, Bethesda, MD, USA) as well as an in-house written code for the GLCM analysis. The mean values of parameters were calculated for each patient. The results demonstrated that the malignant cells from the HT-associated PTC specimens showed lower chromatin fractal dimension (p = 0.0321) and higher lacunarity (p = 0.0038) compared with the corresponding cells from the HT specimens. Also, there was a statistically significant difference between the investigated specimens, in the contrast, correlation, angular second moment, and homogeneity, of the GLCM corresponding to the visual texture of follicular cell chromatin. The differences in chromatin fractal and GLCM parameters could be integrated with other diagnostic methods for the improved evaluation of distinctive features of the HT-associated PTC versus the HT in cytology and surgical pathology specimens.

Fractal analysis of concurrently prepared latex rubber casts of the bronchial and vascular systems of the human lung

Fractal geometry (FG) is a branch of mathematics that instructively characterizes structural complexity. Branched structures are ubiquitous in both the physical and the biological realms. Fractility has therefore been termed nature's design. The fractal properties of the bronchial (airway) system, the pulmonary artery and the pulmonary vein of the human lung generates large respiratory surface area that is crammed in the lung. Also, it permits the inhaled air to intimately approximate the pulmonary capillary blood across a very thin blood-gas barrier through which gas exchange to occur by diffusion. Here, the bronchial (airway) and vascular systems were simultaneously cast with latex rubber. After corrosion, the bronchial and vascular system casts were physically separated and cleared to expose the branches. The morphogenetic (Weibel's) ordering method was used to categorize the branches on which the diameters and the lengths, as well as the angles of bifurcation, were measured. The fractal dimensions (DF) were determined by plotting the total branch measurements against the mean branch diameters on double logarithmic coordinates (axes). The diameter-determined DF values were 2.714 for the bronchial system, 2.882 for the pulmonary artery and 2.334 for the pulmonary vein while the respective values from lengths were 3.098, 3.916 and 4.041. The diameters yielded DF values that were consistent with the properties of fractal structures (i.e. self-similarity and space-filling). The data obtained here compellingly suggest that the design of the bronchial system, the pulmonary artery and the pulmonary vein of the human lung functionally comply with the Hess-Murray law or 'the principle of minimum work'.

The Complexity and Fractal Geometry of Nuclear Medicine Images

Irregularity in shape and behavior is the main feature of every anatomical system, including human organs, tissues, cells, and sub-cellular entities. It has been shown that this property cannot be quantified by means of the classical Euclidean geometry, which is only able to describe regular geometrical objects. In contrast, fractal geometry has been widely applied in several scientific fields. This rapid growth has also produced substantial insights in the biomedical imaging. Consequently, particular attention has been given to the identification of pathognomonic patterns of "shape" in anatomical entities and their changes from natural to pathological states. Despite the advantages of fractal mathematics and several studies demonstrating its applicability to oncological research, many researchers and clinicians remain unaware of its potential. Therefore, this review aims to summarize the complexity and fractal geometry of nuclear medicine images.

Reduced-order modeling of hemodynamics across macroscopic through mesoscopic circulation scales

We propose a hemodynamic reduced-order model bridging macroscopic and mesoscopic blood flow circulation scales from arteries to capillaries. In silico tree-like vascular geometries, mathematically described by graphs, are synthetically generated by means of stochastic growth algorithms constrained by statistical morphological and topological principles. Scale-specific pruning gradation of the tree is then proposed in order to fit computational budget requirement. Different compliant structural models with respect to pressure loads are used depending on vessel walls thicknesses and structures, which vary considerably from macroscopic to mesoscopic circulation scales. Nonlinear rheological properties of blood are also included, and microcirculation network responses are computed for different rheologies. Numerical results are in very good agreement with available experimental measurements. The computational model captures the dynamic transition between large- to small-scale flow pulsatility speeds and magnitudes and wall shear stresses, which have wide-ranging physiological influences.

The cerebral angiome: High resolution MicroCT imaging of the whole brain cerebrovasculature in female and male mice

The cerebrovascular system provides crucial functions that maintain metabolic and homeostatic states of the brain. Despite its integral role of supporting cerebral viability, the topological organization of these networks remains largely uncharacterized. This void in our knowledge surmises entirely from current technological limitations that prevent the capturing of data through the entire depth of the brain. We report high-resolution reconstruction and analysis of the complete vascular network of the entire brain at the capillary level in adult female and male mice using a vascular corrosion cast procedure. Vascular network analysis of the whole brain revealed sex-related differences of vessel hierarchy. In addition, region-specific network analysis demonstrated different patterns of angioarchitecture between brain subregions and sex. Furthermore, our group is the first to provide a three-dimensional analysis of the angioarchitecture and network organization in a single reconstructed tomographic data set that encompasses all hierarchy of vessels in the brain of the adult mouse.

A healthy dose of chaos: Using fractal frameworks for engineering higher-fidelity biomedical systems

Optimal levels of chaos and fractality are distinctly associated with physiological health and function in natural systems. Chaos is a type of nonlinear dynamics that tends to exhibit seemingly random structures, whereas fractality is a measure of the extent of organization underlying such structures. Growing bodies of work are demonstrating both the importance of chaotic dynamics for proper function of natural systems, as well as the suitability of fractal mathematics for characterizing these systems. Here, we review how measures of fractality that quantify the dose of chaos may reflect the state of health across various biological systems, including: brain, skeletal muscle, eyes and vision, lungs, kidneys, tumours, cell regulation, skin and wound repair, bone, vasculature, and the heart. We compare how reports of either too little or too much chaos and fractal complexity can be damaging to normal biological function, and suggest that aiming for the healthy dose of chaos may be an effective strategy for various biomedical applications. We also discuss rising examples of the implementation of fractal theory in designing novel materials, biomedical devices, diagnostics, and clinical therapies. Finally, we explain important mathematical concepts of fractals and chaos, such as fractal dimension, criticality, bifurcation, and iteration, and how they are related to biology. Overall, we promote the effectiveness of fractals in characterizing natural systems, and suggest moving towards using fractal frameworks as a basis for the research and development of better tools for the future of biomedical engineering.

3D Printable Vascular Networks Generated by Accelerated Constrained Constructive Optimization for Tissue Engineering

One of the greatest challenges in fabricating artificial tissues and organs is the incorporation of vascular networks to support the biological requirements of the embedded cells, encouraging tissue formation and maturation. With the advent of 3D printing technology, significant progress has been made with respect to generating vascularized artificial tissues. Current algorithms to generate arterial/venous trees are computationally expensive and offer limited freedom to optimize the resulting structures. Furthermore, there is no method for algorithmic generation of vascular networks that can recapitulate the complexity of the native vasculature for more than two trees, and export directly to a 3D printing format. Here, we report such a method, using an accelerated constructive constrained optimization approach, by decomposing the process into construction, optimization, and collision resolution stages. The new approach reduces computation time to minutes at problem sizes where previous implementations have reported days. With the optimality criterion of maximizing the volume of useful tissue which could be grown around such a network, an approach of alternating stages of construction and batch optimization of all node positions is introduced and shown to yield consistently more optimal networks. The approach does not place a limit on the number of interpenetrating networks that can be constructed in a given space; indeed we demonstrate a biomimetic, liver-like tissue model. Methods to account for the limitations of 3D printing are provided, notably the minimum feature size and infill at sharp angles, through padding and angle reduction, respectively.

Bioprinting Vasculature: Materials, Cells and Emergent Techniques

Despite the great advances that the tissue engineering field has experienced over the last two decades, the amount of in vitro engineered tissues that have reached a stage of clinical trial is limited. While many challenges are still to be overcome, the lack of vascularization represents a major milestone if tissues bigger than approximately 200 µm are to be transplanted. Cell survival and homeostasis is to a large extent conditioned by the oxygen and nutrient transport (as well as waste removal) by blood vessels on their proximity and spontaneous vascularization in vivo is a relatively slow process, leading all together to necrosis of implanted tissues. Thus, in vitro vascularization appears to be a requirement for the advancement of the field. One of the main approaches to this end is the formation of vascular templates that will develop in vitro together with the targeted engineered tissue. Bioprinting, a fast and reliable method for the deposition of cells and materials on a precise manner, appears as an excellent fabrication technique. In this review, we provide a comprehensive background to the fields of vascularization and bioprinting, providing details on the current strategies, cell sources, materials and outcomes of these studies.

Glioblastoma multiforme restructures the topological connectivity of cerebrovascular networks

Glioblastoma multiforme alters healthy tissue vasculature by inducing angiogenesis and vascular remodeling. To fully comprehend the structural and functional properties of the resulting vascular network, it needs to be studied collectively by considering both geometric and topological properties. Utilizing Single Plane Illumination Microscopy (SPIM), the detailed capillary structure in entire healthy and tumor-bearing mouse brains could be resolved in three dimensions. At the scale of the smallest capillaries, the entire vascular systems of bulk U87- and GL261-glioblastoma xenografts, their respective cores, and healthy brain hemispheres were modeled as complex networks and quantified with fundamental topological measures. All individual vessel segments were further quantified geometrically and modular clusters were uncovered and characterized as meta-networks, facilitating an analysis of large-scale connectivity. An inclusive comparison of large tissue sections revealed that geometric properties of individual vessels were altered in glioblastoma in a relatively subtle way, with high intra- and inter-tumor heterogeneity, compared to the impact on the vessel connectivity. A network topology analysis revealed a clear decomposition of large modular structures and hierarchical network organization, while preserving most fundamental topological classifications, in both tumor models with distinct growth patterns. These results augment our understanding of cerebrovascular networks and offer a topological assessment of glioma-induced vascular remodeling. The findings may help understand the emergence of hypoxia and necrosis, and prove valuable for therapeutic interventions such as radiation or antiangiogenic therapy.

A novel porous media-based approach to outflow boundary resistances of 1D arterial blood flow models

In this paper we introduce a novel method for prescribing terminal boundary conditions in one-dimensional arterial flow networks. This is carried out by coupling the terminal arterial vessel with a poro-elastic tube, representing the flow resistance offered by microcirculation. The performance of the proposed porous media-based model has been investigated through several different numerical examples. First, we investigate model parameters that have a profound influence on the flow and pressure distributions of the system. The simulation results have been compared against the waveforms generated by three elements (RCR) Windkessel model. The proposed model is also integrated into a realistic arterial tree, and the results obtained have been compared against experimental data at different locations of the network. The accuracy and simplicity of the proposed model demonstrates that it can be an excellent alternative for the existing models.

Differences in Chromatin Texture and Nuclear Fractal Dimension Between Hashimoto's and Lymphocytic Thyroiditis Lymphocytes

Previous evidence suggested that lymphocytic thyroiditis (LT) was a variant of Hashimoto's thyroiditis (HT), thus the aim of the current study is to quantify structural changes in histological specimens taken from HT and LT patients. A total of 600 images containing a single lymphocyte nucleus (300 nuclei per group) were obtained from 20 patients with HT and LT. In order to quantify changes in the nuclear architecture of investigated lymphocytes, the fractal dimension (FD) and some gray-level co-occurrence matrix texture parameters (angular second moment, inverse difference moment, contrast, entropy, and correlation) were calculated for each nucleus. A statistically significant difference in the FD of the "binary-outlined" nucleus and that of the corresponding "black-and-white" nucleus was detected between HT and LT lymphocyte nuclei. In addition, there was also a statistically significant difference in contrast and correlation between HT and LT lymphocyte nuclei. In conclusion, the results of this study suggested that there was a difference in structural complexity between investigated lymphocyte nuclei; additionally, LT lymphocytes possessed probably more complex texture and larger variations as well as more asymmetrical nuclei compared with HT lymphocytes. Accordingly, these findings indicate that LT is probably not a variant of HT; however, more complex studies are necessary to estimate differences between these types of thyroiditis.

Biofabrication strategies for creating microvascular complexity

Design and fabrication of effective biomimetic vasculatures constitutes a relevant and yet unsolved challenge, lying at the heart of tissue repair and regeneration strategies. Even if cell growth is achieved in 3D tissue scaffolds or advanced implants, tissue viability inevitably requires vascularization, as diffusion can only transport nutrients and eliminate debris within a few hundred microns. This engineered vasculature may need to mimic the intricate branching geometry of native microvasculature, referred to herein as vascular complexity, to efficiently deliver blood and recreate critical interactions between the vascular and perivascular cells as well as parenchymal tissues. This review first describes the importance of vascular complexity in labs- and organs-on-chips, the biomechanical and biochemical signals needed to create and maintain a complex vasculature, and the limitations of current 2D, 2.5D, and 3D culture systems in recreating vascular complexity. We then critically review available strategies for design and biofabrication of complex vasculatures in cell culture platforms, labs- and organs-on-chips, and tissue engineering scaffolds, highlighting their advantages and disadvantages. Finally, challenges and future directions are outlined with the hope of inspiring researchers to create the reliable, efficient and sustainable tools needed for design and biofabrication of complex vasculatures.

Brain Capillary Networks Across Species: A few Simple Organizational Requirements Are Sufficient to Reproduce Both Structure and Function

Despite the key role of the capillaries in neurovascular function, a thorough characterization of cerebral capillary network properties is currently lacking. Here, we define a range of metrics (geometrical, topological, flow, mass transfer, and robustness) for quantification of structural differences between brain areas, organs, species, or patient populations and, in parallel, digitally generate synthetic networks that replicate the key organizational features of anatomical networks (isotropy, connectedness, space-filling nature, convexity of tissue domains, characteristic size). To reach these objectives, we first construct a database of the defined metrics for healthy capillary networks obtained from imaging of mouse and human brains. Results show that anatomical networks are topologically equivalent between the two species and that geometrical metrics only differ in scaling. Based on these results, we then devise a method which employs constrained Voronoi diagrams to generate 3D model synthetic cerebral capillary networks that are locally randomized but homogeneous at the network-scale. With appropriate choice of scaling, these networks have equivalent properties to the anatomical data, demonstrated by comparison of the defined metrics. The ability to synthetically replicate cerebral capillary networks opens a broad range of applications, ranging from systematic computational studies of structure-function relationships in healthy capillary networks to detailed analysis of pathological structural degeneration, or even to the development of templates for fabrication of 3D biomimetic vascular networks embedded in tissue-engineered constructs.

Simulations of blood as a suspension predicts a depth dependent hematocrit in the circulation throughout the cerebral cortex

Recent advances in modeling oxygen supply to cortical brain tissue have begun to elucidate the functional mechanisms of neurovascular coupling. While the principal mechanisms of blood flow regulation after neuronal firing are generally known, mechanistic hemodynamic simulations cannot yet pinpoint the exact spatial and temporal coordination between the network of arteries, arterioles, capillaries and veins for the entire brain. Because of the potential significance of blood flow and oxygen supply simulations for illuminating spatiotemporal regulation inside the cortical microanatomy, there is a need to create mathematical models of the entire cerebral circulation with realistic anatomical detail. Our hypothesis is that an anatomically accurate reconstruction of the cerebrocirculatory architecture will inform about possible regulatory mechanisms of the neurovascular interface. In this article, we introduce large-scale networks of the murine cerebral circulation spanning the Circle of Willis, main cerebral arteries connected to the pial network down to the microcirculation in the capillary bed. Several multiscale models were generated from state-of-the-art neuroimaging data. Using a vascular network construction algorithm, the entire circulation of the middle cerebral artery was synthesized. Blood flow simulations indicate a consistent trend of higher hematocrit in deeper cortical layers, while surface layers with shorter vascular path lengths seem to carry comparatively lower red blood cell (RBC) concentrations. Moreover, the variability of RBC flux decreases with cortical depth. These results support the notion that plasma skimming serves a self-regulating function for maintaining uniform oxygen perfusion to neurons irrespective of their location in the blood supply hierarchy. Our computations also demonstrate the practicality of simulating blood flow for large portions of the mouse brain with existing computer resources. The efficient simulation of blood flow throughout the entire middle cerebral artery (MCA) territory is a promising milestone towards the final aim of predicting blood flow patterns for the entire brain.

The brain’s astonishing cognitive capacity depends on the coordination between neurons and the cerebral circulation, a system known as the neurovascular unit. The spatial and temporal coupling between these two networks is the object of intense research. However, the concise anatomical description of the cerebral circulation has so far been intractable. This paper introduces a methodology for the in silico creation of realistic models for the entire cerebral circulation. This innovation incorporates topological data from several neuroimaging modalities covering three lengths scales as input into a computer algorithm, which assembles anatomically accurate circulatory networks. When simulating blood flow as red blood cells suspended in plasma for experimental and synthetic cortical network models, we discovered that red blood cells tend to be more concentrated in deeper layers of the cortex compared to the surface. RBC fluxes are more homogenous in deeper layers. The phenomenon of depth dependent red blood cell supply supports the notion that the intricate architecture of the cortical microcirculation serves a self-regulating function to maintain uniform oxygen perfusion to neurons. We also demonstrate the practicality of predicting blood flow patterns for the entire brain with existing computer power.

The Relation Between Capillary Transit Times and Hemoglobin Saturation Heterogeneity. Part 2: Capillary Networks

Brain metabolism is highly dependent on continuous oxygen supply. Cortical microvascular networks exhibit heterogeneous blood flow, leading to non-uniform tissue oxygenation and capillary hemoglobin saturation. We recently proposed capillary outflow saturation heterogeneity (COSH) to represent effects of heterogeneity on oxygen supply to tissue regions most vulnerable to hypoxia, and showed that diffusive oxygen exchange among red blood cells within capillaries and among capillaries (diffusive interaction) significantly reduces COSH in simplified geometrical configurations. Here, numerical simulations of oxygen transport in capillary network geometries derived from mouse somatosensory cortex are presented. Diffusive interaction was found to reduce COSH by 41 to 62% compared to simulations where diffusive interaction was excluded. Hemoglobin saturation drop across the microvascular network is strongly correlated with red blood cell transit time, but the coefficient of variation of saturation drop is approximately one third lower. Unexpectedly, the radius of the tissue cylinder supplied by a capillary correlates weakly with the anatomical tissue cylinder radius, but strongly with hemoglobin saturation. Thus, diffusive interaction contributes greatly to the microcirculation's ability to achieve tissue oxygenation, despite heterogeneous capillary transit time and hematocrit distribution. These findings provide insight into the effects of cerebral small vessel disease on tissue oxygenation and brain function.

Investigation of microvascular morphological measures for skeletal muscle tissue oxygenation by image-based modelling in three dimensions

The supply of oxygen in sufficient quantity is vital for the correct functioning of all organs in the human body, especially for skeletal muscle during exercise. Traditionally, microvascular oxygen supply capability is assessed by the analysis of morphological measures on transverse cross-sections of muscle, e.g. capillary density or capillary-to-fibre ratio. In this work, we investigate the relationship between microvascular structure and muscle tissue oxygenation in mice. Phase contrast imaging was performed using synchrotron radiation computed tomography (SR CT) to visualize red blood cells (RBCs) within the microvasculature in mouse soleus muscle. Image-based mathematical modelling of the oxygen diffusion from the RBCs into the muscle tissue was subsequently performed, as well as a morphometric analysis of the microvasculature. The mean tissue oxygenation was then compared with the morphological measures of the microvasculature. RBC volume fraction and spacing (mean distance of any point in tissue to the closest RBC) emerged as the best predictors for muscle tissue oxygenation, followed by length density (summed RBC length over muscle volume). The two-dimensional measures of capillary density and capillary-to-fibre ratio ranked last. We, therefore, conclude that, in order to assess the states of health of muscle tissue, it is advisable to rely on three-dimensional morphological measures rather than on the traditional two-dimensional measures.

The Relation Between Capillary Transit Times and Hemoglobin Saturation Heterogeneity. Part 1: Theoretical Models

Capillary dysfunction impairs oxygen supply to parenchymal cells and often occurs in Alzheimer's disease, diabetes and aging. Disturbed capillary flow patterns have been shown to limit the efficacy of oxygen extraction and can be quantified using capillary transit time heterogeneity (CTH). However, the transit time of red blood cells (RBCs) through the microvasculature is not a direct measure of their capacity for oxygen delivery. Here we examine the relation between CTH and capillary outflow saturation heterogeneity (COSH), which is the heterogeneity of blood oxygen content at the venous end of capillaries. Models for the evolution of hemoglobin saturation heterogeneity (HSH) in capillary networks were developed and validated using a computational model with moving RBCs. Two representative situations were selected: a Krogh cylinder geometry with heterogeneous hemoglobin saturation (HS) at the inflow, and a parallel array of four capillaries. The heterogeneity of HS after converging capillary bifurcations was found to exponentially decrease with a time scale of 0.15-0.21 s due to diffusive interaction between RBCs. Similarly, the HS difference between parallel capillaries also drops exponentially with a time scale of 0.12-0.19 s. These decay times are substantially smaller than measured RBC transit times and only weakly depend on the distance between microvessels. This work shows that diffusive interaction strongly reduces COSH on a small spatial scale. Therefore, we conclude that CTH influences COSH yet does not determine it. The second part of this study will focus on simulations in microvascular networks from the rodent cerebral cortex. Actual estimates of COSH and CTH will then be given.

Mathematical modeling of atherosclerotic plaque destabilization: Role of neovascularization and intraplaque hemorrhage

Observational studies have identified angiogenesis from the adventitial vasa vasorum and intraplaque hemorrhage (IPH) as critical factors in atherosclerotic plaque progression and destabilization. Here we propose a mathematical model incorporating intraplaque neovascularization and hemodynamic calculation with plaque destabilization for the quantitative evaluation of the role of neoangiogenesis and IPH in the vulnerable atherosclerotic plaque formation. An angiogenic microvasculature is generated by two-dimensional nine-point discretization of endothelial cell proliferation and migration from the vasa vasorum. Three key cells (endothelial cells, smooth muscle cells and macrophages) and three key chemicals (vascular endothelial growth factors, extracellular matrix and matrix metalloproteinase) are involved in the plaque progression model, and described by the reaction-diffusion partial differential equations. The hemodynamic calculation of the microcirculation on the generated microvessel network is carried out by coupling the intravascular, interstitial and transvascular flow. The plasma concentration in the interstitial domain is defined as the description of IPH area according to the diffusion and convection with the interstitial fluid flow, as well as the extravascular movement across the leaky vessel wall. The simulation results demonstrate a series of pathophysiological phenomena during the vulnerable progression of an atherosclerotic plaque, including the expanding necrotic core, the exacerbated inflammation, the high microvessel density (MVD) region at the shoulder areas, the transvascular flow through the capillary wall and the IPH. The important role of IPH in the plaque destabilization is evidenced by simulations with varied model parameters. It is found that the IPH can significantly speed up the plaque vulnerability by increasing necrotic core and thinning fibrous cap. In addition, the decreased MVD and vessel permeability may slow down the process of plaque destabilization by reducing the IPH dramatically. We envision that the present model and its future advances can serve as a valuable theoretical platform for studying the dynamic changes in the microenvironment during the plaque destabilization.

Graph analysis of cell clusters forming vascular networks

This manuscript describes the experimental observation of vasculogenesis in chick embryos by means of network analysis. The formation of the vascular network was observed in the area opaca of embryos from 40 to 55 h of development. In the area opaca endothelial cell clusters self-organize as a primitive and approximately regular network of capillaries. The process was observed by bright-field microscopy in control embryos and in embryos treated with Bevacizumab (Avastin^®), an antibody that inhibits the signalling of the vascular endothelial growth factor (VEGF). The sequence of images of the vascular growth were thresholded, and used to quantify the forming network in control and Avastin-treated embryos. This characterization is made by measuring vessels density, number of cell clusters and the largest cluster density. From the original images, the topology of the vascular network was extracted and characterized by means of the usual network metrics such as: the degree distribution, average clustering coefficient, average short path length and assortativity, among others. This analysis allows to monitor how the largest connected cluster of the vascular network evolves in time and provides with quantitative evidence of the disruptive effects that Avastin has on the tree structure of vascular networks.