Oxygen delivery from the cerebral microvasculature to tissue is governed by a single time constant of approximately 6 seconds

Parameter quantification for oxygen transport in the human brain

BACKGROUND AND OBJECTIVE

Oxygen is carried to the brain by blood flow through generations of vessels across a wide range of length scales. This multi-scale nature of blood flow and oxygen transport poses challenges on investigating the mechanisms underlying both healthy and pathological states through imaging techniques alone. Recently, multi-scale models describing whole brain perfusion and oxygen transport have been developed. Such models rely on effective parameters that represent the microscopic properties. While parameters of the perfusion models have been characterised, those for oxygen transport are still lacking. In this study, we set to quantify the parameters associated with oxygen transport and their uncertainties.

METHODS

Effective parameter values of a continuum-based porous multi-scale, multi-compartment oxygen transport model are systematically estimated. In particular, geometric parameters that capture the microvascular topologies are obtained through statistically accurate capillary networks. Maximum consumption rates of oxygen are optimised to uniquely define the oxygen distribution over depth. Simulations are then carried out within a one-dimensional tissue column and a three-dimensional patient-specific brain mesh using the finite element method.

RESULTS

Effective values of the geometric parameters, vessel volume fraction and surface area to volume ratio, are found to be 1.42% and 627 [mm2/mm3], respectively. These values compare well with those acquired from human and monkey vascular samples. Simulation results of the one-dimensional tissue column show qualitative agreement with experimental measurements of tissue oxygen partial pressure in rats. Differences between the oxygenation level in the tissue column and the brain mesh are observed, which highlights the importance of anatomical accuracy. Finally, one-at-a-time sensitivity analysis reveals that the oxygen model is not sensitive to most of its parameters; however, perturbations in oxygen solubilities and plasma to whole blood oxygen concentration ratio have a considerable impact on the tissue oxygenation.

CONCLUSIONS

The findings of this study demonstrate the validity of using a porous continuum approach to model organ-scale oxygen transport and draw attention to the significance of anatomy and parameters associated with inter-compartment diffusion.

Dynamic cerebral autoregulation is governed by two time constants: Arterial transit time and feedback time constant

Dynamic cerebral autoregulation (dCA) is the mechanism that describes how the brain maintains cerebral blood flow approximately constant in response to short-term changes in arterial blood pressure. This is known to be impaired in many different pathological conditions, including ischaemic and haemorrhagic stroke, dementia and traumatic brain injury. Many different approaches have thus been used both to analyse and to quantify this mechanism in a range of healthy and diseased subjects, including data-driven models (in both the time and the frequency domain) and biophysical models. However, despite the substantial body of work on both biophysical models and data-driven models of dCA, there remains little work that links the two together. One of the reasons for this is proposed to be the discrepancies between the time constants that govern dCA in models and in experimental data. In this study, the processes that govern dCA are examined and it is proposed that the application of biophysical models remains limited due to a lack of understanding about the physical processes that are being modelled, partly due to the specific model formulation that has been most widely used (the equivalent electrical circuit). Based on the analysis presented here, it is proposed that the two most important time constants are arterial transit time and feedback time constant. It is therefore time to revisit equivalent electrical circuit models of dCA and to develop a more physiologically realistic alternative, one that can more easily be related to experimental data. KEY POINTS: Dynamic cerebral autoregulation is governed by two time constants. The first time constant is the arterial transit time, rather than the traditional 'RC' time constant widely used in previous models. This arterial transit time is approximately 1 s in the brain. The second time constant is the feedback time constant, which is less accurately known, although it is somewhat larger than the arterial transit time. The equivalent electrical circuit model of dynamic cerebral autoregulation should be replaced with a more physiologically representative model.

The conducted vascular response as a mediator of hypercapnic cerebrovascular reactivity: A modelling study

It is well established that the cerebral blood flow (CBF) shows exquisite sensitivity to changes in the arterial blood partial pressure of CO2 ( [Formula: see text] ), which is reflected by an index termed cerebrovascular reactivity. In response to elevations in [Formula: see text] (hypercapnia), the vessels of the cerebral microvasculature dilate, thereby decreasing the vascular resistance and increasing CBF. Due to the challenges of access, scale and complexity encountered when studying the microvasculature, however, the mechanisms behind cerebrovascular reactivity are not fully understood. Experiments have previously established that the cholinergic release of the Acetylcholine (ACh) neurotransmitter in the cortex is a prerequisite for the hypercapnic response. It is also known that ACh functions as an endothelial-dependent agonist, in which the local administration of ACh elicits local hyperpolarization in the vascular wall; this hyperpolarization signal is then propagated upstream the vascular network through the endothelial layer and is coupled to a vasodilatory response in the vascular smooth muscle (VSM) layer in what is known as the conducted vascular response (CVR). Finally, experimental data indicate that the hypercapnic response is more strongly correlated with the CO2 levels in the tissue than in the arterioles. Accordingly, we hypothesize that the CVR, evoked by increases in local tissue CO2 levels and a subsequent local release of ACh, is responsible for the CBF increase observed in response to elevations in [Formula: see text] . By constructing physiologically grounded dynamic models of CBF and control in the cerebral vasculature, ones that integrate the available knowledge and experimental data, we build a new model of the series of signalling events and pathways underpinning the hypercapnic response, and use the model to provide compelling evidence that corroborates the aforementioned hypothesis. If the CVR indeed acts as a mediator of the hypercapnic response, the proposed mechanism would provide an important addition to our understanding of the repertoire of metabolic feedback mechanisms possessed by the brain and would motivate further in-vivo investigation. We also model the interaction of the hypercapnic response with dynamic cerebral autoregulation (dCA), the collection of mechanisms that the brain possesses to maintain near constant CBF despite perturbations in pressure, and show how the dCA mechanisms, which otherwise tend to be overlooked when analysing experimental results of cerebrovascular reactivity, could play a significant role in shaping the CBF response to elevations in [Formula: see text] . Such in-silico models can be used in tandem with in-vivo experiments to expand our understanding of cerebrovascular diseases, which continue to be among the leading causes of morbidity and mortality in humans.

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 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.

Investigating the role of pericytes in cerebral autoregulation: a modeling study

The brain's inability to store nutrients for more than a few seconds makes it one of the most tightly regulated systems in the body. Driven by metabolic demand, cerebral autoregulation (CA) ensures a constant cerebral blood flow (CBF) over a ±50% change in arterial blood pressure (ABP) from baseline. Recent evidence suggests that pericytes, contractile cells in the capillary bed, play a previously-ignored regulatory role. To elucidate the CA phenomenon, the role of oxygen metabolism, pericyte activity and neural signaling in CBF modulation were quantified. Driven by nutrient metabolism in the tissue and pressure sensitivity in the vasculature, the model introduced here successfully replicates CA. To highlight the role of different vessel sizes, vessels with a diameter above 1 mm were represented using a lumped parameter model while the microvasculature was illustrated as a branching tree network model. This novel approach elucidated the relationship between the microvasculature's nutrient supply and arterial regulation. Capillary responses to local increases in neuronal activity were experimentally determined, showing that pericytes can increase the diameter of the adjacent vessel by 2.5% in approximately 1 s. Their response was quantified and included in the computational model as an active component of the capillary bed. To compare the efficacy model presented here to existing ones, four feedback mechanisms were tested. To simulate dynamic CBF regulation a 10% increase in ABP was imposed. This resulted in a 23.79%-34.33% peak increase in CBF, depending on the nature of the feedback mechanism of the model. The four feedback mechanisms that were studied significantly differ in the response time, ultimately highlighting that capillaries play a fundamental role in the rapid regulation of CBF. Conclusively, this study indicates that while pericytes do not greatly alter the peak CBF change, they play a fundamental role in the speed of regulation.

Thrombus growth modelling and stenosis prediction in the cerebral microvasculature

Cerebral microvascular occlusions cause restriction of blood supply to the brain, thus potentially severely impacting cognitive abilities. Thus, accurate prediction of thrombus growth in realistic geometries is important. Thrombi growth in an existing 13-generation cerebral microvasculature network is simulated here to study the haemodynamic effects of single and multiple blockages on the occlusion of the network. Compared to a single vessel, in a network, the occlusion probability is found to be different. It is the downstream/smaller arterioles (i.e. the 3rd, 4th, 5th, 6th generation arterioles in this study) that tend to reach occlusion first in a network and thus are the critical vessels. Simulations of simultaneous growth of two independent thrombi in the network (referred to here as the two-block case) show a close coupling between the locations of the various blocks in the network, each influencing the other's growth. The presence of the lead block (LB) slows the growth of the trailing block (TB). In some cases, it stops the TB's growth thereby preventing it from occluding the vessel. Findings in this work thus indicate that, to prevent ischaemia, blocks in the smaller arterioles need to be identified and treated first, and that this is more critical if the number of simultaneous blocks is higher.

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.