Mathematical synthesis of the cortical circulation for the whole mouse brain-part I. theory and image integration

The pulsing brain: state of the art and an interdisciplinary perspective

Understanding the pulsing dynamics of tissue and fluids in the intracranial environment is an evolving research theme aimed at gaining new insights into brain physiology and disease progression. This article provides an overview of related research in magnetic resonance imaging, ultrasound medical diagnostics and mathematical modelling of biological tissues and fluids. It highlights recent developments, illustrates current research goals and emphasizes the importance of collaboration between these fields.

Modeling of Blood Flow Dynamics in Rat Somatosensory Cortex

Background: The cerebral microvasculature forms a dense network of interconnected blood vessels where flow is modulated partly by astrocytes. Increased neuronal activity stimulates astrocytes to release vasoactive substances at the endfeet, altering the diameters of connected vessels. Methods: Our study simulated the coupling between blood flow variations and vessel diameter changes driven by astrocytic activity in the rat somatosensory cortex. We developed a framework with three key components: coupling between the vasculature and synthesized astrocytic morphologies, a fluid dynamics model to compute flow in each vascular segment, and a stochastic process replicating the effect of astrocytic endfeet on vessel radii. Results: The model was validated against experimental flow values from the literature across cortical depths. We found that local vasodilation from astrocyte activity increased blood flow, especially in capillaries, exhibiting a layer-specific response in deeper cortical layers. Additionally, the highest blood flow variability occurred in capillaries, emphasizing their role in cerebral perfusion regulation. We discovered that astrocytic activity impacted blood flow dynamics in a localized, clustered manner, with most vascular segments influenced by two to three neighboring endfeet. Conclusions: These insights enhance our understanding of neurovascular coupling and guide future research on blood flow-related diseases.

Mesoscale Brain Mapping: Bridging Scales and Modalities in Neuroimaging - A Symposium Review

Advances in the spatiotemporal resolution and field-of-view of neuroimaging tools are driving mesoscale studies for translational neuroscience. On October 10, 2023, the Center for Mesoscale Mapping (CMM) at the Massachusetts General Hospital (MGH) Athinoula A. Martinos Center for Biomedical Imaging and the Massachusetts Institute of Technology (MIT) Health Sciences Technology based Neuroimaging Training Program (NTP) hosted a symposium exploring the state-of-the-art in this rapidly growing area of research. "Mesoscale Brain Mapping: Bridging Scales and Modalities in Neuroimaging" brought together researchers who use a broad range of imaging techniques to study brain structure and function at the convergence of the microscopic and macroscopic scales. The day-long event centered on areas in which the CMM has established expertise, including the development of emerging technologies and their application to clinical translational needs and basic neuroscience questions. The in-person symposium welcomed more than 150 attendees, including 57 faculty members, 61 postdoctoral fellows, 35 students, and four industry professionals, who represented institutions at the local, regional, and international levels. The symposium also served the training goals of both the CMM and the NTP. The event content, organization, and format were planned collaboratively by the faculty and trainees. Many CMM faculty presented or participated in a panel discussion, thus contributing to the dissemination of both the technologies they have developed under the auspices of the CMM and the findings they have obtained using those technologies. NTP trainees who benefited from the symposium included those who helped to organize the symposium and/or presented posters and gave "flash" oral presentations. In addition to gaining experience from presenting their work, they had opportunities throughout the day to engage in one-on-one discussions with visiting scientists and other faculty, potentially opening the door to future collaborations. The symposium presentations provided a deep exploration of the many technological advances enabling progress in structural and functional mesoscale brain imaging. Finally, students worked closely with the presenting faculty to develop this report summarizing the content of the symposium and putting it in the broader context of the current state of the field to share with the scientific community. We note that the references cited here include conference abstracts corresponding to the symposium poster presentations.

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.

Vascular synthesis based on hemodynamic efficiency principle recapitulates measured cerebral circulation properties in the human brain

Quantifying anatomical and hemodynamical properties of the brain vasculature in vivo is difficult due to limited spatiotemporal resolution neuroimaging, variability between subjects, and bias between acquisition techniques. This work introduces a metabolically inspired vascular synthesis algorithm for creating a digital representation of the cortical blood supply in humans. Spatial organization and segment resistances of a cortical vascular network were generated. Cortical folding and macroscale arterial and venous vessels were reconstructed from anatomical MRI and MR angiography. The remaining network, including ensembles representing the parenchymal capillary bed, were synthesized following a mechanistic principle based on hydrodynamic efficiency of the cortical blood supply. We evaluated the digital model by comparing its simulated values with in vivo healthy human brain measurements of macrovessel blood velocity from phase contrast MRI and capillary bed transit times and bolus arrival times from dynamic susceptibility contrast. We find that measured and simulated values reasonably agree and that relevant neuroimaging observables can be recapitulated in silico. This work provides a basis for describing and testing quantitative aspects of the cerebrovascular circulation that are not directly observable. Future applications of such digital brains include the investigation of the organ-wide effects of simulated vascular and metabolic pathologies.

Cerebral perfusion simulation using realistically generated synthetic trees for healthy and stroke patients

BACKGROUND AND OBJECTIVE

Cerebral vascular diseases are among the most burdensome diseases faced by society. However, investigating the pathophysiology of diseases as well as developing future treatments still relies heavily on expensive in-vivo and in-vitro studies. The generation of realistic, patient-specific models of the cerebrovascular system capable of simulating hemodynamics and perfusion promises the ability to simulate diseased states, therefore accelerating development cycles using in silico studies and opening opportunities for the individual assessment of diseased states, treatment planning, and the prediction of outcomes. By providing a patient-specific, anatomically detailed and validated model of the human cerebral vascular system, we aim to provide the basis for future in silico investigations of the cerebral physiology and pathology.

METHODS

In this retrospective study, a processing pipeline for patient-specific quantification of cerebral perfusion was developed and applied to healthy individuals and a stroke patient. Major arteries are segmented from 3T MR angiography data. A synthetic tree generation algorithm titled tissue-growth based optimization (GBO)1 is used to extend vascular trees beyond the imaging resolution. To investigate the anatomical accuracy of the generated trees, morphological parameters are compared against those of 7 T MRI, 9.4 T MRI, and dissection data. Using the generated vessel model, hemodynamics and perfusion are simulated by solving one-dimensional blood flow equations combined with Darcy flow equations.

RESULTS

Morphological data of three healthy individuals (mean age 47 years ± 15.9 [SD], 2 female) was analyzed. Bifurcation and physiological characteristics of the synthetically generated vessels are comparable to those of dissection data. The inability of MRI based segmentation to resolve small branches and the small volume investigated cause a mismatch in the comparison to MRI data. Cerebral perfusion was estimated for healthy individuals and a stroke patient. The simulated perfusion is compared against Arterial-Spin-Labeling MRI perfusion data. Good qualitative agreement is found between simulated and measured cerebral blood flow (CBF)2. Ischemic regions are predicted well, however ischemia severity is overestimated.

CONCLUSIONS

GBO successfully generates detailed cerebral vascular models with realistic morphological parameters. Simulations based on the resulting networks predict perfusion territories and ischemic regions successfully.

A Deep Learning Approach for Improving Two-Photon Vascular Imaging Speeds

A potential method for tracking neurovascular disease progression over time in preclinical models is multiphoton fluorescence microscopy (MPM), which can image cerebral vasculature with capillary-level resolution. However, obtaining high-quality, three-dimensional images with traditional point scanning MPM is time-consuming and limits sample sizes for chronic studies. Here, we present a convolutional neural network-based (PSSR Res-U-Net architecture) algorithm for fast upscaling of low-resolution or sparsely sampled images and combine it with a segmentation-less vectorization process for 3D reconstruction and statistical analysis of vascular network structure. In doing so, we also demonstrate that the use of semi-synthetic training data can replace the expensive and arduous process of acquiring low- and high-resolution training pairs without compromising vectorization outcomes, and thus open the possibility of utilizing such approaches for other MPM tasks where collecting training data is challenging. We applied our approach to images with large fields of view from a mouse model and show that our method generalizes across imaging depths, disease states and other differences in neurovasculature. Our pretrained models and lightweight architecture can be used to reduce MPM imaging time by up to fourfold without any changes in underlying hardware, thereby enabling deployability across a range of settings.

Sensitivity analysis of a multi-physics model for the vascular microenvironment

The vascular microenvironment is the scale at which microvascular transport, interstitial tissue properties and cell metabolism interact. The vascular microenvironment has been widely studied by means of quantitative approaches, including multi-physics mathematical models as it is a central system for the pathophysiology of many diseases, such as cancer. The microvascular architecture is a key factor for fluid balance and mass transfer in the vascular microenvironment, together with the physical parameters characterizing the vascular wall and the interstitial tissue. The scientific literature of this field has witnessed a long debate about which factor of this multifaceted system is the most relevant. The purpose of this work is to combine the interpretative power of an advanced multi-physics model of the vascular microenvironment with state of the art and robust sensitivity analysis methods, in order to determine the factors that most significantly impact quantities of interest, related in particular to cancer treatment. We are particularly interested in comparing the factors related to the microvascular architecture with the ones affecting the physics of microvascular transport. Ultimately, this work will provide further insight into how the vascular microenvironment affects cancer therapies, such as chemotherapy, radiotherapy or immunotherapy.

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.

Toward an integrative neurovascular framework for studying brain networks

Brain functional connectivity based on the measure of blood oxygen level-dependent (BOLD) functional magnetic resonance imaging (fMRI) signals has become one of the most widely used measurements in human neuroimaging. However, the nature of the functional networks revealed by BOLD fMRI can be ambiguous, as highlighted by a recent series of experiments that have suggested that typical resting-state networks can be replicated from purely vascular or physiologically driven BOLD signals. After going through a brief review of the key concepts of brain network analysis, we explore how the vascular and neuronal systems interact to give rise to the brain functional networks measured with BOLD fMRI. This leads us to emphasize a view of the vascular network not only as a confounding element in fMRI but also as a functionally relevant system that is entangled with the neuronal network. To study the vascular and neuronal underpinnings of BOLD functional connectivity, we consider a combination of methodological avenues based on multiscale and multimodal optical imaging in mice, used in combination with computational models that allow the integration of vascular information to explain functional connectivity.

A network model of glymphatic flow under different experimentally-motivated parametric scenarios

Flow of cerebrospinal fluid (CSF) through perivascular spaces (PVSs) in the brain delivers nutrients, clears metabolic waste, and causes edema formation. Brain-wide imaging cannot resolve PVSs, and high-resolution methods cannot access deep tissue. However, theoretical models provide valuable insight. We model the CSF pathway as a network of hydraulic resistances, using published parameter values. A few parameters (permeability of PVSs and the parenchyma, and dimensions of PVSs and astrocyte endfoot gaps) have wide uncertainties, so we focus on the limits of their ranges by analyzing different parametric scenarios. We identify low-resistance PVSs and high-resistance parenchyma as the only scenario that satisfies three essential criteria: that the flow be driven by a small pressure drop, exhibit good CSF perfusion throughout the cortex, and exhibit a substantial increase in flow during sleep. Our results point to the most important parameters, such as astrocyte endfoot gap dimensions, to be measured in future experiments.

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We model the CSF pathway as a network of hydraulic resistances

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Predictions are bracketed by analyzing parametric scenarios for unknown parameters

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Low-resistance PVSs and high-resistance parenchyma produce realistic flows

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Astrocyte endfoot gap size is among the important parameters to be measured

Control of low flow regions in the cortical vasculature determines optimal arterio-venous ratios

The energy demands of neurons are met by a constant supply of glucose and oxygen via the cerebral vasculature. The cerebral cortex is perfused by dense, parallel arterioles and venules, consistently in imbalanced ratios. Whether and how arteriole-venule arrangement and ratio affect the efficiency of energy delivery to the cortex has remained an unanswered question. Here, we show by mathematical modeling and analysis of the mapped mouse sensory cortex that the perfusive efficiency of the network is predicted to be limited by low-flow regions produced between pairs of arterioles or pairs of venules. Increasing either arteriole or venule density decreases the size of these low-flow regions, but increases their number, setting an optimal ratio between arterioles and venules that matches closely that observed across mammalian cortical vasculature. Low-flow regions are reshaped in complex ways by changes in vascular conductance, creating geometric challenges for matching cortical perfusion with neuronal activity.

A Mesoscale Computational Model for Microvascular Oxygen Transfer

We address a mathematical model for oxygen transfer in the microcirculation. The model includes blood flow and hematocrit transport coupled with the interstitial flow, oxygen transport in the blood and the tissue, including capillary-tissue exchange effects. Moreover, the model is suited to handle arbitrarily complex vascular geometries. The purpose of this study is the validation of the model with respect to classical solutions and the further demonstration of its adequacy to describe the heterogeneity of oxygenation in the tissue microenvironment. Finally, we discuss the importance of these effects in the treatment of cancer using radiotherapy.

Mathematical synthesis of the cortical circulation for the whole mouse brain-part II: Microcirculatory closure

Recent advancements in multiphoton imaging and vascular reconstruction algorithms have increased the amount of data on cerebrovascular circulation for statistical analysis and hemodynamic simulations. Experimental observations offer fundamental insights into capillary network topology but mainly within a narrow field of view typically spanning a small fraction of the cortical surface (less than 2%). In contrast, larger-resolution imaging modalities, such as computed tomography (CT) or magnetic resonance imaging (MRI), have whole-brain coverage but capture only larger blood vessels, overlooking the microscopic capillary bed. To integrate data acquired at multiple length scales with different neuroimaging modalities and to reconcile brain-wide macroscale information with microscale multiphoton data, we developed a method for synthesizing hemodynamically equivalent vascular networks for the entire cerebral circulation. This computational approach is intended to aid in the quantification of patterns of cerebral blood flow and metabolism for the entire brain. In part I, we described the mathematical framework for image-guided generation of synthetic vascular networks covering the large cerebral arteries from the circle of Willis through the pial surface network leading back to the venous sinuses. Here in part II, we introduce novel procedures for creating microcirculatory closure that mimics a realistic capillary bed. We demonstrate our capability to synthesize synthetic vascular networks whose morphometrics match empirical network graphs from three independent state-of-the-art imaging laboratories using different image acquisition and reconstruction protocols. We also successfully synthesized twelve vascular networks of a complete mouse brain hemisphere suitable for performing whole-brain blood flow simulations. Synthetic arterial and venous networks with microvascular closure allow whole-brain hemodynamic predictions. Simulations across all length scales will potentially illuminate organ-wide supply and metabolic functions that are inaccessible to models reconstructed from image data with limited spatial coverage.

Voxelized simulation of cerebral oxygen perfusion elucidates hypoxia in aged mouse cortex

Departures of normal blood flow and metabolite distribution from the cerebral microvasculature into neuronal tissue have been implicated with age-related neurodegeneration. Mathematical models informed by spatially and temporally distributed neuroimage data are becoming instrumental for reconstructing a coherent picture of normal and pathological oxygen delivery throughout the brain. Unfortunately, current mathematical models of cerebral blood flow and oxygen exchange become excessively large in size. They further suffer from boundary effects due to incomplete or physiologically inaccurate computational domains, numerical instabilities due to enormous length scale differences, and convergence problems associated with condition number deterioration at fine mesh resolutions. Our proposed simple finite volume discretization scheme for blood and oxygen microperfusion simulations does not require expensive mesh generation leading to the critical benefit that it drastically reduces matrix size and bandwidth of the coupled oxygen transfer problem. The compact problem formulation yields rapid and stable convergence. Moreover, boundary effects can effectively be suppressed by generating very large replica of the cortical microcirculation in silico using an image-based cerebrovascular network synthesis algorithm, so that boundaries of the perfusion simulations are far removed from the regions of interest. Massive simulations over sizeable portions of the cortex with feature resolution down to the micron scale become tractable with even modest computer resources. The feasibility and accuracy of the novel method is demonstrated and validated with in vivo oxygen perfusion data in cohorts of young and aged mice. Our oxygen exchange simulations quantify steep gradients near penetrating blood vessels and point towards pathological changes that might cause neurodegeneration in aged brains. This research aims to explain mechanistic interactions between anatomical structures and how they might change in diseases or with age. Rigorous quantification of age-related changes is of significant interest because it might aide in the search for imaging biomarkers for dementia and Alzheimer’s disease.

Brain function critically depends on the maintenance of physiological blood supply and metabolism in the cortex. Disturbances to adequate perfusion have been linked to age-related neurodegeneration. However, the precise correlation between age-related hemodynamic changes and the resulting decline in oxygen delivery is not well understood and has not been quantified. Therefore, we introduce a new compact, and therefore highly scalable, computational method for predicting the physiological relationship between hemodynamics and cortical oxygen perfusion for large sections of the cortical microcirculation. We demonstrate the novel mesh generation-free (MGF), multi-scale simulation approach through realistic in vivo case studies of cortical microperfusion in the mouse brain. We further validate mechanistic correlations and a quantitative relationship between blood flow and brain oxygenation using experimental data from cohorts of young, middle aged and old mouse brains. Our computational approach overcomes size and performance limitations of previous unstructured meshing techniques to enable the prediction of oxygen tension with a spatial resolution of least two orders of magnitude higher than previously possible. Our simulation results support the hypothesis that structural changes in the microvasculature induce hypoxic pockets in the aged brain that are absent in the healthy, young mouse.

Laplacian Flow Dynamics on Geometric Graphs for Anatomical Modeling of Cerebrovascular Networks

Generating computational anatomical models of cerebrovascular networks is vital for improving clinical practice and understanding brain oxygen transport. This is achieved by extracting graph-based representations based on pre-mapping of vascular structures. Recent graphing methods can provide smooth vessels trajectories and well-connected vascular topology. However, they require water-tight surface meshes as inputs. Furthermore, adding vessels radii information on their graph compartments restricts their alignment along vascular centerlines. Here, we propose a novel graphing scheme that works with relaxed input requirements and intrinsically captures vessel radii information. The proposed approach is based on deforming geometric graphs constructed within vascular boundaries. Under a laplacian optimization framework, we assign affinity weights on the initial geometry that drives its iterative contraction toward vessels centerlines. We present a mechanism to decimate graph structure at each run and a convergence criterion to stop the process. A refinement technique is then introduced to obtain final vascular models. Our implementation is available on https://github.com/Damseh/VascularGraph. We benchmarked our results with that obtained using other efficient and state-of-the-art graphing schemes, validating on both synthetic and real angiograms acquired with different imaging modalities. The experiments indicate that the proposed scheme produces the lowest geometric and topological error rates on various angiograms. Furthermore, it surpasses other techniques in providing representative models that capture all anatomical aspects of vascular structures.

Computational feasibility of simulating changes in blood flow through whole-organ vascular networks from radiation injury

Vasculature is necessary to the healthy function of most tissues. In radiation therapy, injury of the vasculature can have both beneficial and detrimental effects, such as tumor starvation, cardiac fibrosis, and white-matter necrosis. These effects are caused by changes in blood flow due to the vascular injury. Previously, research has focused on simulating the radiation injury of vasculature in small volumes of tissue, ignoring the systemic effects of local damage on blood flow. Little is known about the computational feasibility of simulating the radiation injury to whole-organ vascular networks. The goal of this study was to test the computational feasibility of simulating the dose deposition to a whole-organ vascular network and the resulting change in blood flow. To do this, we developed an amorphous track-structure model to transport radiation and combined this with existing methods to model the vasculature and blood flow rates. We assessed the algorithm's computational scalability, execution time, and memory usage. The data demonstrated it is computationally feasible to calculate the radiation dose and resulting changes in blood flow from 2 million protons to a network comprising 8.5 billion blood vessels (approximately the number in the human brain) in 87 hours using a 128-node cluster. Furthermore, the algorithm demonstrated both strong and weak scalability, meaning that additional computational resources can reduce the execution time further. These results demonstrate, for the first time, that it is computationally feasible to calculate radiation dose deposition in whole-organ vascular networks. These findings provide key insights into the computational aspects of modeling whole-organ radiation damage. Modeling the effects radiation has on vasculature could prove useful in the study of radiation effects on tissues, organs, and organisms.

Computational feasibility of simulating whole-organ vascular networks

The human body contains approximately 20 billion blood vessels, which transport nutrients, oxygen, immune cells, and signals throughout the body. The brain's vasculature includes up to 9 billion of these vessels to support cognition, motor processes, and myriad other vital functions. To model blood flowing through a vasculature, a geometric description of the vessels is required. Previously reported attempts to model vascular geometries have produced highly-detailed models. These models, however, are limited to a small fraction of the human brain, and little was known about the feasibility of computationally modeling whole-organ-sized networks. We implemented a fractal-based algorithm to construct a vasculature the size of the human brain and evaluated the algorithm's speed and memory requirements. Using high-performance computing systems, the algorithm constructed a vasculature comprising 17 billion vessels in 1960 core-hours, or 49 minutes of wall-clock time, and required less than 32 GB of memory per node. We demonstrated strong scalability that was limited mainly by input/output operations. The results of this study demonstrated, for the first time, that it is feasible to computationally model the vasculature of the whole human brain. These findings provide key insights into the computational aspects of modeling whole-organ vasculature.

Computational feasibility of calculating the steady-state blood flow rate through the vasculature of the entire human body

The human body contains approximately 20 billion individual blood vessels that deliver nutrients and oxygen to tissues. While blood flow is a well-developed field of research, no previous studies have calculated the blood flow rates through more than 5 million connected vessels. The goal of this study was to test if it is computationally feasible to calculate the blood flow rates through a vasculature equal in size to that of the human body. We designed and implemented a two-step algorithm to calculate the blood flow rates using principles of steady-state fluid dynamics. Steady-state fluid dynamics is an accurate approximation for the microvascular and venous structures in the human body. To determine the computational feasibility, we measured and evaluated the execution time, scalability, and memory usage to quantify the computational requirements. We demonstrated that it is computationally feasible to calculate the blood flow rate through 17 billion vessels in 6.5 hours using 256 compute nodes. The computational modeling of blood flow rate in entire organisms may find application in research on drug delivery, treatment of cancer metastases, and modulation of physiological performance.