Cerebral microcirculation and oxygen tension in the human secondary cortex

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.

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.

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.

Tissue-Engineered Microvessels: A Review of Current Engineering Strategies and Applications

Microvessels, including arterioles, capillaries, and venules, play an important role in regulating blood flow, enabling nutrient and waste exchange, and facilitating immune surveillance. Due to their important roles in maintaining normal function in human tissues, a substantial effort has been devoted to developing tissue-engineered models to study endothelium-related biology and pathology. Various engineering strategies have been developed to recapitulate the structural, cellular, and molecular hallmarks of native human microvessels in vitro. In this review, recent progress in engineering approaches, key components, and culture platforms for tissue-engineered human microvessel models is summarized. Then, tissue-specific models, and the major applications of tissue-engineered microvessels in development, disease modeling, drug screening and delivery, and vascularization in tissue engineering, are reviewed. Finally, future research directions for the field are discussed.

Linking Vascular Structure and Function: Image-Based Virtual Populations of the Retina

Purpose

This study explored the relationship among microvascular parameters as delineated by optical coherence tomography angiography (OCTA) and retinal perfusion. Here, we introduce a versatile framework to examine the interplay between the retinal vascular structure and function by generating virtual vasculatures from central retinal vessels to macular capillaries. Also, we have developed a hemodynamics model that evaluates the associations between vascular morphology and retinal perfusion.

Methods

The generation of the vasculature is based on the distribution of four clinical parameters pertaining to the dimension and blood pressure of the central retinal vessels, constructive constrained optimization, and Voronoi diagrams. Arterial and venous trees are generated in the temporal retina and connected through three layers of capillaries at different depths in the macula. The correlations between total retinal blood flow and macular flow fraction and vascular morphology are derived as Spearman rank coefficients, and uncertainty from input parameters is quantified.

Results

A virtual cohort of 200 healthy vasculatures was generated. Means and standard deviations for retinal blood flow and macular flow fraction were 20.80 ± 7.86 µL/min and 15.04% ± 5.42%, respectively. Retinal blood flow was correlated with vessel area density, vessel diameter index, fractal dimension, and vessel caliber index. The macular flow fraction was not correlated with any morphological metrics.

Conclusions

The proposed framework is able to reproduce vascular networks in the macula that are morphologically and functionally similar to real vasculature. The framework provides quantitative insights into how macular perfusion can be affected by changes in vascular morphology delineated on OCTA.

Modeling cerebrovascular responses to assess the impact of the collateral circulation following middle cerebral artery occlusion

OBJECTIVE

An improved understanding of the role of the leptomeningeal collateral circulation in blood flow compensation following middle cerebral artery (MCA) occlusion can contribute to more effective treatment development for ischemic stroke. The present study introduces a model of the cerebral circulation to predict cerebral blood flow and tissue oxygenation following MCA occlusion.

METHODS

The model incorporates flow regulation mechanisms based on changes in pressure, shear stress, and metabolic demand. Oxygen saturation in cerebral vessels and tissue is calculated using a Krogh cylinder model. The model is used to assess the effects of changes in oxygen demand and arterial pressure on cerebral blood flow and oxygenation after MCA occlusion.

RESULTS

An increase from five to 11 leptomeningeal collateral vessels was shown to increase the oxygen saturation in the region distal to the occlusion by nearly 100%. Post-occlusion, the model also predicted a loss of autoregulation and a decrease in flow to the ischemic territory as oxygen demand was increased; these results were consistent with data from experiments that induced cerebral ischemia.

CONCLUSIONS

This study highlights the importance of leptomeningeal collaterals following MCA occlusion and reinforces the idea that lower oxygen demand and higher arterial pressure improve conditions of flow and oxygenation.

Simulating Cerebral Edema and Ischemia After Traumatic Acute Subdural Hematoma Using Triphasic Swelling Biomechanics

Poor outcome following traumatic acute subdural hematoma (ASDH) is associated with the severity of the primary injury and secondary injury including cerebral edema and ischemia. However, the underlying secondary injury mechanism contributing to elevated intracranial pressure (ICP) and high mortality rate remains unclear. Cerebral edema occurs in response to the exposure of the intracellular fixed charge density (FCD) after cell death, causing ICP to increase. The increased ICP from swollen tissue compresses blood vessels in adjacent tissue, restricting blood flow and leading to ischemic damage. We hypothesize that the mass occupying effect of ASDH exacerbates the ischemic injury, leading to ICP elevation, which is an indicator of high mortality rate in the clinic. Using FEBio (febio.org) and triphasic swelling biomechanics, this study modeled clinically relevant ASDHs and simulated post-traumatic brain swelling and ischemia to predict ICP. Results showed that common convexity ASDH significantly increased ICP by exacerbating ischemic injury, and surgical removal of the convexity ASDH may control ICP by preventing ischemia progression. However, in cases where the primary injury is very severe, surgical intervention alone may not effectively decrease ICP, as the contribution of the hematoma to the elevated ICP is insignificant. In addition, interhemispheric ASDH, located between the cerebral hemispheres, does not significantly exacerbate ischemia, supporting the conservative surgical management generally recommended for interhemispheric ASDH. The joint effect of the mass occupying effect of the blood clot and resulting ischemia contributes to elevated ICP which may increase mortality. Our novel approach may improve the fidelity of predicting patient outcome after motor vehicle crashes and traumatic brain injuries due to other causes.

Neuromonitoring after Pediatric Cardiac Arrest: Cerebral Physiology and Injury Stratification

BACKGROUND

Significant long-term neurologic disability occurs in survivors of pediatric cardiac arrest, primarily due to hypoxic-ischemic brain injury. Postresuscitation care focuses on preventing secondary injury and the pathophysiologic cascade that leads to neuronal cell death. These injury processes include reperfusion injury, perturbations in cerebral blood flow, disturbed oxygen metabolism, impaired autoregulation, cerebral edema, and hyperthermia. Postresuscitation care also focuses on early injury stratification to allow clinicians to identify patients who could benefit from neuroprotective interventions in clinical trials and enable targeted therapeutics.

METHODS

In this review, we provide an overview of postcardiac arrest pathophysiology, explore the role of neuromonitoring in understanding postcardiac arrest cerebral physiology, and summarize the evidence supporting the use of neuromonitoring devices to guide pediatric postcardiac arrest care. We provide an in-depth review of the neuromonitoring modalities that measure cerebral perfusion, oxygenation, and function, as well as neuroimaging, serum biomarkers, and the implications of targeted temperature management.

RESULTS

For each modality, we provide an in-depth review of its impact on treatment, its ability to stratify hypoxic-ischemic brain injury severity, and its role in neuroprognostication.

CONCLUSION

Potential therapeutic targets and future directions are discussed, with the hope that multimodality monitoring can shift postarrest care from a one-size-fits-all model to an individualized model that uses cerebrovascular physiology to reduce secondary brain injury, increase accuracy of neuroprognostication, and improve outcomes.

Approaches to vascular network, blood flow, and metabolite distribution modeling in brain tissue

The cardiovascular system plays a key role in the transport of nutrients, ensuring a continuous supply of all cells of the body with the metabolites necessary for life. The blood supply to the brain is carried out by the large arteries located on its surface, which branch into smaller arterioles that penetrate the cerebral cortex and feed the capillary bed, thereby forming an extensive branching network. The formation of blood vessels is carried out via vasculogenesis and angiogenesis, which play an important role in both embryo and adult life. The review presents approaches to modeling various aspects of both the formation of vascular networks and the construction of the formed arterial tree. In addition, a brief description of models that allows one to study the blood flow in various parts of the circulatory system and the spatiotemporal metabolite distribution in brain tissues is given. Experimental study of these issues is not always possible due to both the complexity of the cardiovascular system and the mechanisms through which the perfusion of all body cells is carried out. In this regard, mathematical models are a good tool for studying hemodynamics and can be used in clinical practice to diagnose vascular diseases and assess the need for treatment.

An anatomically detailed arterial-venous network model. Cerebral and coronary circulation

In recent years, several works have addressed the problem of modeling blood flow phenomena in veins, as a response to increasing interest in modeling pathological conditions occurring in the venous network and their connection with the rest of the circulatory system. In this context, one-dimensional models have proven to be extremely efficient in delivering predictions in agreement with in-vivo observations. Pursuing the increase of anatomical accuracy and its connection to physiological principles in haemodynamics simulations, the main aim of this work is to describe a novel closed-loop Anatomically-Detailed Arterial-Venous Network (ADAVN) model. An extremely refined description of the arterial network consisting of 2,185 arterial vessels is coupled to a novel venous network featuring high level of anatomical detail in cerebral and coronary vascular territories. The entire venous network comprises 189 venous vessels, 79 of which drain the brain and 14 are coronary veins. Fundamental physiological mechanisms accounting for the interaction of brain blood flow with the cerebro-spinal fluid and of the coronary circulation with the cardiac mechanics are considered. Several issues related to the coupling of arterial and venous vessels at the microcirculation level are discussed in detail. Numerical simulations are compared to patient records published in the literature to show the descriptive capabilities of the model. Furthermore, a local sensitivity analysis is performed, evidencing the high impact of the venous circulation on main cardiovascular variables.

Multi-level optical angiography for photodynamic therapy

Blood flow imaging is widely applied in photodynamic therapy (PDT) to provide vascular morphological and statistical parameters. This approach relies on the intensity of time-domain signal differences between blood vessels and background tissues; therefore, it often ignores differences within the vasculature and cannot accommodate abundant structural information. This study proposes a multi-level optical angiography (MOA) method for PDT. It can enhance capillaries and image vessels at different levels by measuring the signal frequency shift associated with red blood cell motion. The experimental results regarding the PDT-induced chorioallantoic membrane model showed that the proposed method could not only perform multi-level angiography but also provide more accurate quantitative information regarding various vascular parameters. This MOA method has potential applications in PDT studies.

MESH-FREE HIGH-RESOLUTION SIMULATION OF CEREBROCORTICAL OXYGEN SUPPLY WITH FAST FOURIER PRECONDITIONING

Oxygen transfer from blood vessels to cortical brain tissue is representative of a class of problems with mixed-domain character. Large-scale efficient computation of tissue oxygen concentration is dependent on the manner in which the tubular network of blood vessels is coupled to the tissue mesh. Models which explicitly resolve the interface between the tissue and vasculature with a contiguous mesh are prohibitively expensive for very dense cerebral microvasculature. We propose a mixed-domain mesh-free technique whereby a vascular anatomical network (VAN) represented as a thin directed graph serves for convection of blood oxygen, and the surrounding extravascular tissue is represented as a Cartesian grid of 3D voxels throughout which oxygen is transported by diffusion. We split the network and tissue meshes by the Schur complement method of domain decomposition to obtain a reduced set of system equations for the tissue oxygen concentration. The use of a Cartesian grid allows the corresponding matrix equation to be solved approximately with a fast Fourier transform based Poisson solver, which serves as an effective preconditioner for Krylov subspace iteration. The performance of this method enables the steady state simulation of cortical oxygen perfusion for anatomically accurate vascular networks down to single micron resolution without the need for supercomputers.

Practitioner Points

We present a novel mixed-domain framework for efficiently modeling O ₂ extraction kinetics in the brain. Model equations are generated by graph-theoretic methods for mixed domains.Dual mesh domain decomposition with FFT preconditioning yields very fast simulation times for extremely high spatial resolution.

Quantification of hypoxic regions distant from occlusions in cerebral penetrating arteriole trees

The microvasculature plays a key role in oxygen transport in the mammalian brain. Despite the close coupling between cerebral vascular geometry and local oxygen demand, recent experiments have reported that microvascular occlusions can lead to unexpected distant tissue hypoxia and infarction. To better understand the spatial correlation between the hypoxic regions and the occlusion sites, we used both in vivo experiments and in silico simulations to investigate the effects of occlusions in cerebral penetrating arteriole trees on tissue hypoxia. In a rat model of microembolisation, 25 μm microspheres were injected through the carotid artery to occlude penetrating arterioles. In representative models of human cortical columns, the penetrating arterioles were occluded by simulating the transport of microspheres of the same size and the oxygen transport was simulated using a Green’s function method. The locations of microspheres and hypoxic regions were segmented, and two novel distance analyses were implemented to study their spatial correlation. The distant hypoxic regions were found to be present in both experiments and simulations, and mainly due to the hypoperfusion in the region downstream of the occlusion site. Furthermore, a reasonable agreement for the spatial correlation between hypoxic regions and occlusion sites is shown between experiments and simulations, which indicates the good applicability of in silico models in understanding the response of cerebral blood flow and oxygen transport to microemboli.

The brain function depends on the continuous oxygen supply through the bloodstream inside the microvasculature. Occlusions in the microvascular network will disturb the oxygen delivery in the brain and result in hypoxic tissues that can lead to infarction and cognitive dysfunction. To aid in understanding the formation of hypoxic tissues caused by micro-occlusions in the penetrating arteriole trees, we use rodent experiments and simulations of human vascular networks to study the spatial correlations between the hypoxic regions and the occlusion locations. Our results suggest that hypoxic regions can form distally from the occlusion site, which agrees with the previous observations in the rat brain. These distant hypoxic regions are primarily due to the lack of blood flow in the brain tissues downstream of the occlusion. Moreover, a reasonable agreement of the spatial relationship is found between the experiments and the simulations, which indicates the applicability of in silico models to study the effects of microemboli on the brain tissue.

Mathematical modelling of haemorrhagic transformation within a multi-scale microvasculature network

Objective Haemorrhagic transformation (HT) is one of the most common complications after ischaemic stroke caused by damage to the blood-brain barrier (BBB) that could be the result of stroke progression or a complication of stroke treatment with reperfusion therapy. The aim of this study is to develop further a previous simple HT mathematical model into an enlarged multi-scale microvasculature model in order to investigate the effects of HT on the surrounding tissue and vasculature. In addition, this study investigates the relationship between tissue displacement and vascular geometry. Approach By modelling tissue displacement, capillary compression, hydraulic conductivity in tissue and vascular permeability, we establish a mathematical model to describe the change of intracranial pressure (ICP) surrounding the damaged vascular bed after HT onset applied to a 3D multi-scale microvasculature. The use of a voxel-scale model then enables us to compare our HT simulation with available clinical imaging data for perfusion and cerebral blood volume (CBV) in the multi-scale microvasculature network. Main results We showed that the haematoma diameter and the maximum tissue displacement are approximately proportional to the diameter of the breakdown vessel. Based on the voxel-scale model, we found that perfusion reduces by approximately 13-17 % and CBV reduces by around 20-25 % after HT onset due to the effect of capillary compression caused by increased interstitial pressure. The results are in good agreement with the limited experimental data. Significance This model, by enabling us to bridge the gap between the microvascular scale and clinically measurable parameters, thus provides a foundation for more detailed validation and understanding of HT in patients.

A few upstream bifurcations drive the spatial distribution of red blood cells in model microfluidic networks

The physics of blood flow in small vessel networks is dominated by the interactions between Red Blood Cells (RBCs), plasma and blood vessel walls. The resulting couplings between the microvessel network architecture and the heterogeneous distribution of RBCs at network-scale are still poorly understood. The main goal of this paper is to elucidate how a local effect, such as RBC partitioning at individual bifurcations, interacts with the global structure of the flow field to induce specific preferential locations of RBCs in model microfluidic networks. First, using experimental results, we demonstrate that persistent perturbations to the established hematocrit profile after diverging bifurcations may bias RBC partitioning at the next bifurcations. By performing a sensitivity analysis based upon network models of RBC flow, we show that these perturbations may propagate from bifurcation to bifurcation, leading to an outsized impact of a few crucial upstream bifurcations on the distribution of RBCs at network-scale. Based on measured hematocrit profiles, we further construct a modified RBC partitioning model that accounts for the incomplete relaxation of RBCs at these bifurcations. This model allows us to explain how the flow field results in a single pattern of RBC preferential location in some networks, while it leads to the emergence of two different patterns of RBC preferential location in others. Our findings have important implications in understanding and modeling blood flow in physiological and pathological conditions.

Cerebrospinal fluid dynamics coupled to the global circulation in holistic setting: Mathematical models, numerical methods and applications

This paper presents a mathematical model of the global, arterio-venous circulation in the entire human body, coupled to a refined description of the cerebrospinal fluid (CSF) dynamics in the craniospinal cavity. The present model represents a substantially revised version of the original Müller-Toro mathematical model. It includes one-dimensional (1D), non-linear systems of partial differential equations for 323 major blood vessels and 85 zero-dimensional, differential-algebraic systems for the remaining components. Highlights include the myogenic mechanism of cerebral blood regulation; refined vasculature for the inner ear, the brainstem and the cerebellum; and viscoelastic, rather than purely elastic, models for all blood vessels, arterial and venous. The derived 1D parabolic systems of partial differential equations for all major vessels are approximated by hyperbolic systems with stiff source terms following a relaxation approach. A major novelty of this paper is the coupling of the circulation, as described, to a refined description of the CSF dynamics in the craniospinal cavity, following Linninger et al. The numerical solution methodology employed to approximate the hyperbolic non-linear systems of partial differential equations with stiff source terms is based on the Arbitrary DERivative Riemann problem finite volume framework, supplemented with a well-balanced formulation, and a local time stepping procedure. The full model is validated through comparison of computational results against published data and bespoke MRI measurements. Then we present two medical applications: (i) transverse sinus stenoses and their relation to Idiopathic Intracranial Hypertension; and (ii) extra-cranial venous strictures and their impact in the inner ear circulation, and its implications for Ménière's disease.

Mathematical modelling of haemorrhagic transformation after ischaemic stroke

With an increasingly elderly population globally, the impacts of cerebrovascular diseases, such as stroke and dementia, become increasingly significant. Haemorrhagic transformation (HT) is one of the most common complications of ischaemic stroke that is caused by dysfunction of endothelial cells in the blood-brain barrier (BBB) and that can be exacerbated by thrombolytic therapy. Recent studies also suggest that HT can lead to an increase in intracranial pressure (ICP) and result in capillary compression. The aim of this study is to develop a mathematical model that can be used to simulate the consequence of HT over a range of vasculature length scales. We use a 2D vasculature model to investigate the severity of HT with different vascular geometry. The resulting model shows that the haematoma radius is approximately constant across different length scales (100-1000μm) and in good agreement with the available experimental data. In addition, this study identified that the effects of capillary compression do appear to have a significant impact on the leakage fraction of blood and hence act to restrain the development of a haematoma.

Modelling the effects of cerebral microthrombi on tissue oxygenation and cell death

Thrombectomy, the mechanical removal of a clot, is the most common way to treat ischaemic stroke with large vessel occlusions. However, perfusion cannot always be restored after such an intervention. It has been hypothesised that the absence of reperfusion is at least partially due to the clot fragments that block the downstream vessels. In this paper, we present a new way of quantifying the effects of cerebral microthrombi on oxygen transport to tissue in terms of hypoxia and ischaemia. The oxygen transport was simulated with the Green's function method on physiologically representative microvascular cubes, which was found independent of both microvascular geometry and length scale. The microthrombi occlusions were then simulated in the microvasculature, which were extravasated over time with a new thrombus extravasation model. The tissue hypoxic fraction was fitted as a sigmoidal function of vessel blockage fraction, which was then taken to be a function of time after the formation of microthrombi occlusions. A novel hypoxia-based 3-state cell death model was finally proposed to simulate the hypoxic tissue damage over time. Using the cell death model, the impact of a certain degree of microthrombi occlusions on tissue viability and microinfarct volume can be predicted over time. Quantifying the impact of microthrombi on oxygen transport and tissue death will play an important role in full brain models of ischaemic stroke and thrombectomy.

The Ageing Brain: Investigating the Role of Age in Changes to the Human Cerebral Microvasculature With an in silico Model

Ageing causes extensive structural changes to the human cerebral microvasculature, which have a significant effect on capillary bed perfusion and oxygen transport. Current models of brain capillary networks in the literature focus on healthy adult brains and do not capture the effects of ageing, which is critical when studying neurodegenerative diseases. This study builds upon a statistically accurate model of the human cerebral microvasculature based on ex-vivo morphological data. This model is adapted for “healthy” ageing using in-vivo measurements from mice at three distinct age groups—young, middle-aged, and old. From this new model, blood and molecular exchange parameters are calculated such as permeability and surface-area-to-volume ratio, and compared across the three age groups. The ability to alter the model vessel-by-vessel is used to create a continuous gradient of ageing. It was found that surface-area-to-volume ratio reduced in old age by 6% and permeability by 24% from middle-age to old age, and variability within the networks also increased with age. The ageing gradient indicated a threshold in the ageing process around 75 years old, after which small changes have an amplified effect on blood flow properties. This gradient enables comparison of studies measuring cerebral properties at discrete points in time. The response of middle aged and old aged capillary beds to micro-emboli showed a lower robustness of the old age capillary bed to vessel occlusion. As the brain ages, there is thus increased vulnerability of the microvasculature—with a “tipping point” beyond which further remodeling of the microvasculature has exaggerated effects on the brain. When developing in-silico models of the brain, age is a very important consideration to accurately assess risk factors for cognitive decline and isolate early biomarkers of microvascular health.

Mathematical modeling of cerebral capillary blood flow heterogeneity and its effect on brain tissue oxygen levels

Maintaining cerebral blood flow is critical for adequate neuronal function. Previous computational models of brain capillary networks have predicted that heterogeneous cerebral capillary flow patterns result in lower brain tissue partial oxygen pressures PO2). However, these previous models have often considered simple capillary networks in terms of their geometric properties. In this current work, we developed and analyzed computational models of brain capillary networks to determine how perturbations of network properties impact tissue oxygen levels. The models include variabilities in both their geometric (segment lengths and diameters) and three-dimensional, topological structure. Two classes of capillary network models are considered. The first consists of equations for the oxygen partial pressure, PO₂, in both a capillary network and the surrounding tissue. In order to gain insight into the behavior of this detailed model, we also consider a reduced model for changes in PO₂ in just the capillary network. The main result is that for a general class of networks, random perturbations of either segment diameters or conductances will always, on average, decrease the average tissue oxygen levels. This result is supported through both simulations of the models and mathematical analysis. Our results promise to expand our understanding of cerebral capillary blood flow and its impact on the brain function in health and disease.

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.

Adaptive constrained constructive optimisation for complex vascularisation processes

Mimicking angiogenetic processes in vascular territories acquires importance in the analysis of the multi-scale circulatory cascade and the coupling between blood flow and cell function. The present work extends, in several aspects, the Constrained Constructive Optimisation (CCO) algorithm to tackle complex automatic vascularisation tasks. The main extensions are based on the integration of adaptive optimisation criteria and multi-staged space-filling strategies which enhance the modelling capabilities of CCO for specific vascular architectures. Moreover, this vascular outgrowth can be performed either from scratch or from an existing network of vessels. Hence, the vascular territory is defined as a partition of vascular, avascular and carriage domains (the last one contains vessels but not terminals) allowing one to model complex vascular domains. In turn, the multi-staged space-filling approach allows one to delineate a sequence of biologically-inspired stages during the vascularisation process by exploiting different constraints, optimisation strategies and domain partitions stage by stage, improving the consistency with the architectural hierarchy observed in anatomical structures. With these features, the aDaptive CCO (DCCO) algorithm proposed here aims at improving the modelled network anatomy. The capabilities of the DCCO algorithm are assessed with a number of anatomically realistic scenarios.

The role of vascular complexity on optimal junction exponents

We examine the role of complexity on arterial tree structures, determining globally optimal vessel arrangements using the Simulated AnneaLing Vascular Optimization algorithm, a computational method which we have previously used to reproduce features of cardiac and cerebral vasculatures. In order to progress computational methods for growing arterial networks, deeper understanding of the stability of computational arterial growth algorithms to complexity, variations in physiological parameters (such as metabolic costs for maintaining and pumping blood), and underlying assumptions regarding the value of junction exponents is needed. We determine the globally optimal structure of two-dimensional arterial trees; analysing how physiological parameters affect tree morphology and optimal bifurcation exponent. We find that considering the full complexity of arterial trees is essential for determining the fundamental properties of vasculatures. We conclude that optimisation-based arterial growth algorithms are stable against uncertainties in physiological parameters, while optimal bifurcation exponents (a key parameter for many arterial growth algorithms) are affected by the complexity of vascular networks and the boundary conditions dictated by organs.

Modelling the impact of clot fragmentation on the microcirculation after thrombectomy

Many ischaemic stroke patients who have a mechanical removal of their clot (thrombectomy) do not get reperfusion of tissue despite the thrombus being removed. One hypothesis for this 'no-reperfusion' phenomenon is micro-emboli fragmenting off the large clot during thrombectomy and occluding smaller blood vessels downstream of the clot location. This is impossible to observe in-vivo and so we here develop an in-silico model based on in-vitro experiments to model the effect of micro-emboli on brain tissue. Through in-vitro experiments we obtain, under a variety of clot consistencies and thrombectomy techniques, micro-emboli distributions post-thrombectomy. Blood flow through the microcirculation is modelled for statistically accurate voxels of brain microvasculature including penetrating arterioles and capillary beds. A novel micro-emboli algorithm, informed by the experimental data, is used to simulate the impact of micro-emboli successively entering the penetrating arterioles and the capillary bed. Scaled-up blood flow parameters-permeability and coupling coefficients-are calculated under various conditions. We find that capillary beds are more susceptible to occlusions than the penetrating arterioles with a 4x greater drop in permeability per volume of vessel occluded. Individual microvascular geometries determine robustness to micro-emboli. Hard clot fragmentation leads to larger micro-emboli and larger drops in blood flow for a given number of micro-emboli. Thrombectomy technique has a large impact on clot fragmentation and hence occlusions in the microvasculature. As such, in-silico modelling of mechanical thrombectomy predicts that clot specific factors, interventional technique, and microvascular geometry strongly influence reperfusion of the brain. Micro-emboli are likely contributory to the phenomenon of no-reperfusion following successful removal of a major clot.

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.

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.

Surface area-to-volume ratio, not cellular viscoelasticity, is the major determinant of red blood cell traversal through small channels

The remarkable deformability of red blood cells (RBCs) depends on the viscoelasticity of the plasma membrane and cell contents and the surface area to volume (SA:V) ratio; however, it remains unclear which of these factors is the key determinant for passage through small capillaries. We used a microfluidic device to examine the traversal of normal, stiffened, swollen, parasitised and immature RBCs. We show that dramatic stiffening of RBCs had no measurable effect on their ability to traverse small channels. By contrast, a moderate decrease in the SA:V ratio had a marked effect on the equivalent cylinder diameter that is traversable by RBCs of similar cellular viscoelasticity. We developed a finite element model that provides a coherent rationale for the experimental observations, based on the nonlinear mechanical behaviour of the RBC membrane skeleton. We conclude that the SA:V ratio should be given more prominence in studies of RBC pathologies.

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.

The mass transfer coefficient for oxygen transport from blood to tissue in cerebral cortex

The functioning of cerebral cortex depends on adequate tissue oxygenation. MRI-based techniques allow estimation of blood oxygen levels, tissue perfusion, and oxygen consumption rate (CMRO₂), but do not directly measure partial pressure of oxygen (PO₂) in tissue. To address the estimation of tissue PO₂, the oxygen mass transfer coefficient (KTO₂) is here defined as the CMRO₂ divided by the difference in spatially averaged PO₂ between blood and tissue, and is estimated by analyzing Krogh-cylinder type models. Resistance to radial diffusion of oxygen from microvessels to tissue is distributed within vessels and in the extravascular tissue. The value of KTO₂ is shown to depend strongly on vascular length density and also on microvessel tube hematocrits and diameters, but to be insensitive to blood flow rate and to transient changes in flow or oxygen consumption. Estimated values of KTO₂ are higher than implied by previous studies, implying smaller declines in PO₂ from blood to tissue. Average tissue PO₂ can be estimated from MRI-based measurements as average blood PO₂ minus the product of KTO₂ and CMRO₂. For oxygen consumption rates and vascular densities typical of mouse cortex, the predicted difference between average blood and tissue PO₂ is about 10 mmHg.

Model-based evaluation of the microhemodynamic effects of PEGylated HBOC molecules in the rat brain cortex: a laser speckle imaging study

Hemoglobin-based oxygen carriers (HBOCs) were developed with the aim of substituting transfusions in emergency events. However, they exhibit adverse events, such as nitric oxide (NO) scavenging, vasoactivity, enhanced platelet aggregation, presently hampering their clinical application. The impact of two prototypical PEGylated HBOCs, Euro-PEG-Hb and PEG-HbO₂, endowed by different oxygen affinities and hydrodynamic volumes, was assessed on the cerebrocortical parenchymal microhemodynamics, and extravasation through the blood-brain-barrier (BBB) by laser speckle contrast imaging (LSCI) method and near-infrared (NIR) imaging, respectively. By evaluating voxel-wise cerebrocortical red blood cell velocity, non-invasively for its mean kernel-wise value ( v ¯ RBC ), and model-derived kernel-wise predictions for microregional tissue hematocrit, ^TH_t, and fractional change in hematocrit-corrected vascular resistance, R', as measures of potential adverse effects (enhanced platelet aggregation and vasoactivity, respectively) we found i) no significant difference between tested HBOCs in the systemic and microregional parameters, and in the relative spatial dispersion of ^TH_t, and R' as additional measures of HBOC-related adverse effects, and ii) no extravasation through BBB by Euro-PEG-Hb. We conclude that Euro-PEG-Hb does not exhibit adverse effects in the brain microcirculation that could be directly attributed to NO scavenging.

Multiscale modeling of human cerebrovasculature: A hybrid approach using image-based geometry and a mathematical algorithm

The cerebral vasculature has a complex and hierarchical network, ranging from vessels of a few millimeters to superficial cortical vessels with diameters of a few hundred micrometers, and to the microvasculature (arteriole/venule) and capillary beds in the cortex. In standard imaging techniques, it is difficult to segment all vessels in the network, especially in the case of the human brain. This study proposes a hybrid modeling approach that determines these networks by explicitly segmenting the large vessels from medical images and employing a novel vascular generation algorithm. The framework enables vasculatures to be generated at coarse and fine scales for individual arteries and veins with vascular subregions, following the personalized anatomy of the brain and macroscale vasculatures. In this study, the vascular structures of superficial cortical (pial) vessels before they penetrate the cortex are modeled as a mesoscale vasculature. The validity of the present approach is demonstrated through comparisons with partially observed data from existing measurements of the vessel distributions on the brain surface, pathway fractal features, and vascular territories of the major cerebral arteries. Additionally, this validation provides some biological insights: (i) vascular pathways may form to ensure a reasonable supply of blood to the local surface area; (ii) fractal features of vascular pathways are not sensitive to overall and local brain geometries; and (iii) whole pathways connecting the upstream and downstream entire-scale cerebral circulation are highly dependent on the local curvature of the cerebral sulci.

Cerebral autoregulation in the complex vascular networks of the brain is an amazing achievement. We believe that numerical analysis of the cerebral blood circulation using an anatomically precise vascular model provides a powerful tool for evaluating the direct relationships between local- and global-scale blood flows. However, there is a lack of information about the overall vascular pathways in the human brain, preventing a monolithic model of the human cerebrovasculature from being established. This paper presents a multiscale model of human cerebrovasculature based on a hybrid approach that uses image-based geometries and a newly developed mathematical algorithm. One important argument of this paper is the validity of the cerebrovasculature represented in the model, which reflects anatomical features of major cerebral vasculatures and brain shape, and has strong similarities with available data for human superficial cortical vessels. Investigations of the reconstructed model allow us to derive some biological insights and associated hypotheses for the cerebral vasculature. The authors believe the present cerebrovascular model can be applied to numerical simulations of the entire-scale cerebral blood flow.

A multiscale subvoxel perfusion model to estimate diffusive capillary wall conductivity in multiple sclerosis lesions from perfusion MRI data

We propose a new mathematical model to learn capillary leakage coefficients from dynamic susceptibility contrast MRI data. To this end, we derive an embedded mixed-dimension flow and transport model for brain tissue perfusion on a subvoxel scale. This model is used to obtain the contrast agent concentration distribution in a single MRI voxel during a perfusion MRI sequence. We further present a magnetic resonance signal model for the considered sequence including a model for local susceptibility effects. This allows modeling MR signal-time curves that can be compared with clinical MRI data. The proposed model can be used as a forward model in the inverse modeling problem of inferring model parameters such as the diffusive capillary wall conductivity. Acute multiple sclerosis lesions are associated with a breach in the integrity of the blood-brain barrier. Applying the model to perfusion MR data of a patient with acute multiple sclerosis lesions, we conclude that diffusive capillary wall conductivity is a good indicator for characterizing activity of lesions, even if other patient-specific model parameters are not well-known.

Differential Expression of CD31 and Von Willebrand Factor on Endothelial Cells in Different Regions of the Human Brain: Potential Implications for Cerebral Malaria Pathogenesis

Cerebral microvascular endothelial cells (CMVECs) line the vascular system of the brain and are the chief cells in the formation and function of the blood brain barrier (BBB). These cells are heterogeneous along the cerebral vasculature and any dysfunctional state in these cells can result in a local loss of function of the BBB in any region of the brain. There is currently no report on the distribution and variation of the CMVECs in different brain regions in humans. This study investigated microcirculation in the adult human brain by the characterization of the expression pattern of brain endothelial cell markers in different brain regions. Five different brain regions consisting of the visual cortex, the hippocampus, the precentral gyrus, the postcentral gyrus, and the rhinal cortex obtained from three normal adult human brain specimens were studied and analyzed for the expression of the endothelial cell markers: cluster of differentiation 31 (CD31) and von-Willebrand-Factor (vWF) through immunohistochemistry. We observed differences in the expression pattern of CD31 and vWF between the gray matter and the white matter in the brain regions. Furthermore, there were also regional variations in the pattern of expression of the endothelial cell biomarkers. Thus, this suggests differences in the nature of vascularization in various regions of the human brain. These observations also suggest the existence of variation in structure and function of different brain regions, which could reflect in the pathophysiological outcomes in a diseased state.

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.

More homogeneous capillary flow and oxygenation in deeper cortical layers correlate with increased oxygen extraction

Our understanding of how capillary blood flow and oxygen distribute across cortical layers to meet the local metabolic demand is incomplete. We addressed this question by using two-photon imaging of resting-state microvascular oxygen partial pressure (PO₂) and flow in the whisker barrel cortex in awake mice. Our measurements in layers I-V show that the capillary red-blood-cell flux and oxygenation heterogeneity, and the intracapillary resistance to oxygen delivery, all decrease with depth, reaching a minimum around layer IV, while the depth-dependent oxygen extraction fraction is increased in layer IV, where oxygen demand is presumably the highest. Our findings suggest that more homogeneous distribution of the physiological observables relevant to oxygen transport to tissue is an important part of the microvascular network adaptation to local brain metabolism. These results will inform the biophysical models of layer-specific cerebral oxygen delivery and consumption and improve our understanding of the diseases that affect cerebral microcirculation.

A new framework for assessing subject-specific whole brain circulation and perfusion using MRI-based measurements and a multi-scale continuous flow model

A large variety of severe medical conditions involve alterations in microvascular circulation. Hence, measurements or simulation of circulation and perfusion has considerable clinical value and can be used for diagnostics, evaluation of treatment efficacy, and for surgical planning. However, the accuracy of traditional tracer kinetic one-compartment models is limited due to scale dependency. As a remedy, we propose a scale invariant mathematical framework for simulating whole brain perfusion. The suggested framework is based on a segmentation of anatomical geometry down to imaging voxel resolution. Large vessels in the arterial and venous network are identified from time-of-flight (ToF) and quantitative susceptibility mapping (QSM). Macro-scale flow in the large-vessel-network is accurately modelled using the Hagen-Poiseuille equation, whereas capillary flow is treated as two-compartment porous media flow. Macro-scale flow is coupled with micro-scale flow by a spatially distributing support function in the terminal endings. Perfusion is defined as the transition of fluid from the arterial to the venous compartment. We demonstrate a whole brain simulation of tracer propagation on a realistic geometric model of the human brain, where the model comprises distinct areas of grey and white matter, as well as large vessels in the arterial and venous vascular network. Our proposed framework is an accurate and viable alternative to traditional compartment models, with high relevance for simulation of brain perfusion and also for restoration of field parameters in clinical brain perfusion applications.

An accurate simulation of blood-flow in the human brain can be used for improved diagnostics and assignment of personalized treatment regimes. However, current algorithms are limited to simulation of blood flow within tumours only, and in terms of parameter estimation, traditional compartment models have limited accuracy due to lack of spatial connectivity within the models. As a remedy, we propose a data-driven computational fluid dynamics model where the geometric domains for simulation are defined from state-of-the art MR acquisitions enabling a segmentation of large arteries and veins. In the capillary tissue we apply a two-compartment porous media model, where the perfusion is pressure-driven and is defined as the transition of blood from arterial to venous side. In addition, we propose a model for dealing with the intermediate scale problem where the vessels are undetectable and the flow does not adhere to requirements of porous media flow. For this scale, we propose a support function distributing the fluid in a nearby region around the vessel terminals. Combining these elements, we have developed a novel full human brain blood-flow simulator.

Development of a globally optimised model of the cerebral arteries

The cerebral arteries are difficult to reproduce from first principles, featuring interwoven territories, and intricate layers of grey and white matter with differing metabolic demand. The aim of this study was to identify the ideal configuration of arteries required to sustain an entire brain hemisphere based on minimisation of the energy required to supply the tissue. The 3D distribution of grey and white matter within a healthy human brain was first segmented from magnetic resonance images. A novel simulated annealing algorithm was then applied to determine the optimal configuration of arteries required to supply brain tissue. The model was validated through comparison of this ideal, entirely optimised, brain vasculature with the structure and properties of real arteries. This analysis established that the human cerebral vasculature is highly optimised; closely resembling the most energy efficient arrangement of vessels. In addition to local adherence to fluid dynamical optimisation principles, the optimised vasculature reproduced expected brain perfusion territories, featuring well-defined boundaries between anterior, middle and posterior regions. This validated brain vascular model and algorithm can be used for patient-specific modelling of stroke and cerebral haemodynamics, identification of sub-optimal conditions associated with vascular disease, and optimising vascular structures for tissue engineering applications and artificial organ design.

The need for mathematical modelling of spatial drug distribution within the brain

The blood brain barrier (BBB) is the main barrier that separates the blood from the brain. Because of the BBB, the drug concentration-time profile in the brain may be substantially different from that in the blood. Within the brain, the drug is subject to distributional and elimination processes: diffusion, bulk flow of the brain extracellular fluid (ECF), extra-intracellular exchange, bulk flow of the cerebrospinal fluid (CSF), binding and metabolism. Drug effects are driven by the concentration of a drug at the site of its target and by drug-target interactions. Therefore, a quantitative understanding is needed of the distribution of a drug within the brain in order to predict its effect. Mathematical models can help in the understanding of drug distribution within the brain. The aim of this review is to provide a comprehensive overview of system-specific and drug-specific properties that affect the local distribution of drugs in the brain and of currently existing mathematical models that describe local drug distribution within the brain. Furthermore, we provide an overview on which processes have been addressed in these models and which have not. Altogether, we conclude that there is a need for a more comprehensive and integrated model that fills the current gaps in predicting the local drug distribution within the brain.

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

Microcirculation plays a significant role in cerebral metabolism and blood flow control, yet explaining and predicting functional mechanisms remains elusive because it is difficult to make physiologically accurate mathematical models of the vascular network. As a precursor to the human brain, this paper presents a computational framework for synthesizing anatomically accurate network models for the cortical blood supply in mouse. It addresses two critical deficiencies in cerebrovascular modeling. At the microscopic length scale of individual capillaries, we present a novel synthesis method for building anatomically consistent capillary networks with loops and anastomoses (=microcirculatory closure). This overcomes shortcomings in existing algorithms which are unable to create closed circulatory networks. A second critical innovation allows the incorporation of detailed anatomical features from image data into vascular growth. Specifically, computed tomography and two photon laser scanning microscopy data are input into the novel synthesis algorithm to build the cortical circulation for the entire mouse brain in silico. Computer predictions of blood flow and oxygen exchange executed on synthetic large-scale network models are expected to elucidate poorly understood functional mechanisms of the cerebral circulation.

Spatio-temporal dynamics of cerebral capillary segments with stalling red blood cells

Optical coherence tomography (OCT) allows label-free imaging of red blood cell (RBC) flux within capillaries with high spatio-temporal resolution. In this study, we utilized time-series OCT-angiography to demonstrate interruptions in capillary RBC flux in mouse brain in vivo. We noticed ∼7.5% of ∼200 capillaries had at least one stall in awake mice with chronic windows during a 9-min recording. At any instant, ∼0.45% of capillaries were stalled. Average stall duration was ∼15 s but could last over 1 min. Stalls were more frequent and longer lasting in acute window preparations. Further, isoflurane anesthesia in chronic preparations caused an increase in the number of stalls. In repeated imaging, the same segments had a tendency to stall again over a period of one month. In awake animals, functional stimulation decreased the observance of stalling events. Stalling segments were located distally, away from the first couple of arteriolar-side capillary branches and their average RBC and plasma velocities were lower than nonstalling capillaries within the same region. This first systematic analysis of capillary RBC stalls in the brain, enabled by rapid and continuous volumetric imaging of capillaries with OCT-angiography, will lead to future investigations of the potential role of stalling events in cerebral pathologies.

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.

A computational model for microcirculation including Fahraeus-Lindqvist effect, plasma skimming and fluid exchange with the tissue interstitium

We present a two-phase model for microcirculation that describes the interaction of plasma with red blood cells. The model takes into account of typical effects characterizing the microcirculation, such as the Fahraeus-Lindqvist effect and plasma skimming. Besides these features, the model describes the interaction of capillaries with the surrounding tissue. More precisely, the model accounts for the interaction of capillary transmural flow with the surrounding interstitial pressure. Furthermore, the capillaries are represented as one-dimensional channels with arbitrary, possibly curved configuration. The latter two features rely on the unique ability of the model to account for variations of flow rate and pressure along the axis of the capillary, according to a local differential formulation of mass and momentum conservation. Indeed, the model stands on a solid mathematical foundation, which is also addressed in this work. In particular, we present the model derivation, the variational formulation, and its approximation using the finite element method. Finally, we conclude the work with a comparative computational study of the importance of the Fahraeus-Lindqvist, plasma skimming, and capillary leakage effects on the distribution of flow in a microvascular network.

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.

Modeling Cerebral Blood Flow Dependence on Carbon Dioxide and Mean Arterial Blood Pressure in the Immature Brain With Accounting for the Germinal Matrix

Intraventricular hemorrhage (IVH) is one of the most critical complications in the development of preterm infants. The likelihood of IVH is strongly associated with disturbances in cerebral blood flow (CBF) and with microvascular fragility in the germinal matrix (GM). The CBF value and its reactivity to changes in arterial carbon dioxide pressure (pCO₂) and mean arterial blood pressure (MABP) are relevant indicators in the clinical assessment of preterm infants. The objective of the present study is mathematical modeling of the influence of pCO₂ and MABP on CBF in immature brain, based on clinical data collected from 265 preterm infants with 23–30 gestational weeks. The model was adapted to the peculiarities of immature brain by taking into account the morphological characteristics of the GM capillary network and vascular reactivity, according to gestational and postnatal age. An analysis of model based values of CBF and its reactivity to changes in MABP and pCO₂ was performed separately for each gestational week and for the first two days of life both for preterm infants with and without IVH. The developed model for the estimation of CBF was validated against equivalent experimental measurements taken from the literature. A good agreement between the estimated values of CBF, as well as its reaction on changes in MABP and pCO₂ and the equivalent values obtained in experimental studies was shown.