Mathematical models reveal the key role of cerebral microvasculature in oxygen delivery and tissue health
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, I developed mathematical models of oxygen transport in the cerebral microvasculature. For the first time, we quantified the relationship between the loss of micro-vessels and the brain tissue hypoxia (Xue et al., 2021). We applied the models in in silico trials for the treatment of ischaemic stroke through collaborations with clinicians, biologists and bioengineers from the EU-funded INSIST project (Miller et al., 2021). A follow-up work extended the length scale to a cortical column, where the spatial correlation between occlusions and tissue hypoxic regions were quantified and validated against animal experiments (Xue et al., 2022). The cortical column models have also been used to study the transit time distributions, which reflect underlying microvascular structure and flow characteristics (Payne et al., 2025).
References
2025
Transit time mean and variance are markers of vascular network structure, wall shear stress distribution and oxygen extraction fraction
Perfusion measurements provide information about flow magnitude, but more detailed information is found from transit time distributions (TTD). Whether TTDs provide intrinsic (flow-independent) information about vascular geometry or just flow field remains unknown. We propose a new approach to calculate TTD, based on wall shear stress (WSS). We show that constant WSS yields zero-variance TTD. Simulations in statistical networks show that mean transit time (MTT) and capillary transit time heterogeneity (CTH) are primarily determined by pathway number distribution rather than pressure drop distribution. Using 1000 statistically generated cortical columns, we show that (1) the central volume theorem provides a very good approximation for MTT, hence is a measure of tissue permeability; (2) CTH/MTT ratio, RTH (relative transit time heterogeneity), is a marker of WSS variability; and (3) RTH is inversely related to network oxygen extraction fraction (OEF) but only weakly related to MTT. RTH is below one in animal models, but above one in humans, indicating that WSS distribution is tighter in small animals (lower RTH and higher OEF), due to higher metabolic rate. Human WSS distribution appears to be an inherent property, since simulations show much larger RTH. Finally, WSS distribution is unaffected in ageing, but altered in pathology.
2022
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.
2021
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.
In silico trials for treatment of acute ischemic stroke: design and implementation
An in silico trial simulates a disease and its corresponding therapies on a cohort of virtual patients to support the development and evaluation of medical devices, drugs, and treatment. In silico trials have the potential to refine, reduce cost, and partially replace current in vivo studies, namely clinical trials and animal testing. We present the design and implementation of an in silico trial for treatment of acute ischemic stroke. We propose an event-based modelling approach for the simulation of a disease and injury, where changes to the state of the system (the events) are assumed to be instantaneous. Using this approach we are able to combine a diverse set of models, spanning multiple time scales, to model acute ischemic stroke, treatment, and resulting brain tissue injury. The in silico trial is designed to be modular to aid development and reproducibility. It provides a comprehensive framework for application to any potential in silico trial. A statistical population model is used to generate cohorts of virtual patients. Patient functional outcomes are also predicted with a statistical model, using treatment and injury results and the patient’s clinical parameters. We demonstrate the functionality of the event-based modelling approach and trial framework by running proof of concept in silico trials. The proof of concept trials simulate the same cohort of patients twice: once with successful treatment (successful recanalisation) and once with unsuccessful treatment (unsuccessful treatment). Ways to overcome some of the challenges and difficulties in setting up such an in silico trial are discussed, such as validation and computational limitations.
