The influence of network structure on the transport of blood in the human cerebral microvasculature

Microvascular plasticity in mouse stroke model recovery: Anatomy statistics, dynamics measured by longitudinal in vivo two-photon angiography, network vectorization

This manuscript quantitatively investigates remodeling dynamics of the cortical microvascular network (thousands of connected capillaries) following photothrombotic ischemia (cubic millimeter volume, imaged weekly) using a novel in vivo two-photon angiography and high throughput vascular vectorization method. The results suggest distinct temporal patterns of cerebrovascular plasticity, with acute remodeling peaking at one week post-stroke. The network architecture then gradually stabilizes, returning to a new steady state after four weeks. These findings align with previous literature on neuronal plasticity, highlighting the correlation between neuronal and neurovascular remodeling. Quantitative analysis of neurovascular networks using length- and strand-based statistical measures reveals intricate changes in network anatomy and topology. The distance and strand-length statistics show significant alterations, with a peak of plasticity observed at one week post-stroke, followed by a gradual return to baseline. The orientation statistic plasticity peaks at two weeks, gradually approaching the (conserved across subjects) stroke signature. The underlying mechanism of the vascular response (angiogenesis vs. tissue deformation), however, is yet unexplored. Overall, the combination of chronic two-photon angiography, vascular vectorization, reconstruction/visualization, and statistical analysis enables both qualitative and quantitative assessments of neurovascular remodeling dynamics, demonstrating a method for investigating cortical microvascular network disorders and the therapeutic modes of action thereof.

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.

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.

Microvascular Tissue Engineering-A Review

Tissue engineering and regenerative medicine have come a long way in recent decades, but the lack of functioning vasculature is still a major obstacle preventing the development of thicker, physiologically relevant tissue constructs. A large part of this obstacle lies in the development of the vessels on a microscale-the microvasculature-that are crucial for oxygen and nutrient delivery. In this review, we present the state of the art in the field of microvascular tissue engineering and demonstrate the challenges for future research in various sections of the field. Finally, we illustrate the potential strategies for addressing some of those challenges.

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.

A porous circulation model of the human brain for in silico clinical trials in ischaemic stroke

The advancement of ischaemic stroke treatment relies on resource-intensive experiments and clinical trials. In order to improve ischaemic stroke treatments, such as thrombolysis and thrombectomy, we target the development of computational tools for in silico trials which can partially replace these animal and human experiments with fast simulations. This study proposes a model that will serve as part of a predictive unit within an in silico clinical trial estimating patient outcome as a function of treatment. In particular, the present work aims at the development and evaluation of an organ-scale microcirculation model of the human brain for perfusion prediction. The model relies on a three-compartment porous continuum approach. Firstly, a fast and robust method is established to compute the anisotropic permeability tensors representing arterioles and venules. Secondly, vessel encoded arterial spin labelling magnetic resonance imaging and clustering are employed to create an anatomically accurate mapping between the microcirculation and large arteries by identifying superficial perfusion territories. Thirdly, the parameter space of the problem is reduced by analysing the governing equations and experimental data. Fourthly, a parameter optimization is conducted. Finally, simulations are performed with the tuned model to obtain perfusion maps corresponding to an open and an occluded (ischaemic stroke) scenario. The perfusion map in the occluded vessel scenario shows promising qualitative agreement with computed tomography images of a patient with ischaemic stroke caused by large vessel occlusion. The results highlight that in the case of vessel occlusion (i) identifying perfusion territories is essential to capture the location and extent of underperfused regions and (ii) anisotropic permeability tensors are required to give quantitatively realistic estimation of perfusion change. In the future, the model will be thoroughly validated against experiments.

Manufacturing of poly(ethylene glycol diacrylate)-based hollow microvessels using microfluidics

Biocompatible and self-standing poly(ethylene glycol diacrylate)-based hollow microvessels were fabricated from a microfluidic device using microfluidic principles.

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.

Quantitative histomorphometry of capillary microstructure in deep white matter

White matter lesions represent a major risk factor for dementia in elderly people. Magnetic Resonance Imaging (MRI) studies have demonstrated cerebral blood flow reduction in age-related white matter lesions, indicating that vascular alterations are involved in developing white matter lesions. Hypoperfusion and changes in capillary morphology are generally linked to dementia. However, a quantitative study describing these microvascular alterations in white matter lesions is missing in the literature; most previous microvascular studies being on the cortex. The aim of this work is to identify and quantify capillary microstructural changes involved in the appearance of deep subcortical lesions (DSCL). We characterize the distribution of capillary diameter, thickness, and density in the deep white matter in a population of 75 elderly subjects, stratified into three equal groups according to DSCL: Control (subject without DSCL), Lesion (sample presenting DSCL), and Normal Appearing White Matter (NAWM, the subject presented DSCL but not at the sampled tissue location). Tissue samples were selected from the Cognitive Function and Aging Study (CFAS), a cohort representative of an aging population, from which immunohistochemically-labeled histological images were acquired. To accurately estimate capillary diameters and thicknesses from the 2D histological images, we also introduce a novel semi-automatic method robust to non-perpendicular incidence angle of capillaries into the imaging plane, and to non-circular deformations of capillary cross sections. Subjects with DSCL presented a significant increase in capillary wall thickness, a decrease in the diameter intra-subject variability (but not in the mean), and a decrease in capillary density. No significant difference was observed between controls and NAWM. Both capillary wall thickening and reduction in capillary density contribute to the reduction of cerebral blood flow previously reported for white matter lesions. The obtained distributions provide reliable statistics of capillary microstructure useful to inform the modeling of human cerebral blood flow, for instance to define microcirculation models for their estimation from MRI or to perform realistic cerebral blood flow simulations.

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.

Simultaneous magnetic resonance diffusion and pseudo-diffusion tensor imaging

PURPOSE

An emerging topic in diffusion magnetic resonance is imaging blood microcirculation alongside water diffusion using the intravoxel incoherent motion (IVIM) model. Recently, a combined IVIM diffusion tensor imaging (IVIM-DTI) model was proposed, which accounts for both anisotropic pseudo-diffusion due to blood microcirculation and anisotropic diffusion due to tissue microstructures. In this article, we propose a robust IVIM-DTI approach for simultaneous diffusion and pseudo-diffusion tensor imaging.

METHODS

Conventional IVIM estimation methods can be broadly divided into two-step (diffusion and pseudo-diffusion estimated separately) and one-step (diffusion and pseudo-diffusion estimated simultaneously) methods. Here, both methods were applied on the IVIM-DTI model. An improved one-step method based on damped Gauss-Newton algorithm and a Gaussian prior for the model parameters was also introduced. The sensitivities of these methods to different parameter initializations were tested with realistic in silico simulations and experimental in vivo data.

RESULTS

The one-step damped Gauss-Newton method with a Gaussian prior was less sensitive to noise and the choice of initial parameters and delivered more accurate estimates of IVIM-DTI parameters compared to the other methods.

CONCLUSION

One-step estimation using damped Gauss-Newton and a Gaussian prior is a robust method for simultaneous diffusion and pseudo-diffusion tensor imaging using IVIM-DTI model. Magn Reson Med 79:2367-2378, 2018. © 2017 The Authors Magnetic Resonance in Medicine published by Wiley Periodicals, Inc. on behalf of International Society for Magnetic Resonance in Medicine. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Investigating the effects of a penetrating vessel occlusion with a multi-scale microvasculature model of the human cerebral cortex

The effect of the microvasculature on observed clinical parameters, such as cerebral blood flow, is poorly understood. This is partly due to the gap between the vessels that can be individually imaged in humans and the microvasculature, meaning that mathematical models are required to understand the role of the microvasculature. As a result, a multi-scale model based on morphological data was developed here that is able to model large regions of the human microvasculature. From this model, a clear layering of flow (and 1-dimensional depth profiles) was observed within a voxel, with the flow in the microvasculature being driven predominantly by the geometry of the penetrating vessels. It also appears that the pressure and flow are decoupled, both in healthy vasculatures and in those where occlusions have occurred, again due to the topology of the penetrating vessels shunting flow between them. Occlusion of a penetrating arteriole resulted in a very high degree of overlap of blood pressure drop with experimentally observed cell death. However, drops in blood flow were far more widespread, providing additional support for the theory that pericyte controlled regulation on the capillary scale likely plays a large part in the perfusion of tissue post-occlusion.

A statistical model of the penetrating arterioles and venules in the human cerebral cortex

OBJECTIVE

Models of the cerebral microvasculature are required at many different scales in order to understand the effects of microvascular topology on CBF. There are, however, no data-driven models at the arteriolar/venular scale. In this paper, we develop a data-driven algorithm based on available data to generate statistically accurate penetrating arterioles and venules.

METHODS

A novel order-based density-filling algorithm is developed based on the statistical data including bifurcating angles, LDRs, and area ratios. Three thousand simulations are presented, and the results validated against morphological data. These are combined with a previous capillary network in order to calculate full vascular network parameters.

RESULTS

Statistically accurate penetrating trees were successfully generated. All properties provided a good fit to experimental data. The k exponent had a median of 2.5 and an interquartile range of 1.75-3.7. CBF showed a standard deviation ranging from ±18% to ±34% of the mean, depending on the penetrating vessel diameter.

CONCLUSIONS

Small CBF variations indicate that the topology of the penetrating vessels plays only a small part in the large regional variations of CBF seen in the brain. These results open up the possibility of efficient oxygen and blood flow simulations at MRI voxel scales which can be directly validated against MRI data.

A two-pool model to describe the IVIM cerebral perfusion

IntraVoxel Incoherent Motion (IVIM) is a magnetic resonance imaging (MRI) technique capable of measuring perfusion-related parameters. In this manuscript, we show that the mono-exponential model commonly used to process IVIM data might be challenged, especially at short diffusion times. Eleven rat datasets were acquired at 7T using a diffusion-weighted pulsed gradient spin echo sequence with b-values ranging from 7 to 2500 s/mm² at three diffusion times. The IVIM signals, obtained by removing the diffusion component from the raw MR signal, were fitted to the standard mono-exponential model, a bi-exponential model and the Kennan model. The Akaike information criterion used to find the best model to fit the data demonstrates that, at short diffusion times, the bi-exponential IVIM model is most appropriate. The results obtained by comparing the experimental data to a dictionary of numerical simulations of the IVIM signal in microvascular networks support the hypothesis that such a bi-exponential behavior can be explained by considering the contribution of two vascular pools: capillaries and somewhat larger vessels.

Modelling the effects of cerebral microvasculature morphology on oxygen transport

The cerebral microvasculature plays a vital role in adequately supplying blood to the brain. Determining the health of the cerebral microvasculature is important during pathological conditions, such as stroke and dementia. Recent studies have shown the complex relationship between cerebral metabolic rate and transit time distribution, the transit times of all the possible pathways available dependent on network topology. In this paper, we extend a recently developed technique to solve for residue function, the amount of tracer left in the vasculature at any time, and transit time distribution in an existing model of the cerebral microvasculature to calculate cerebral metabolism. We present the mathematical theory needed to solve for oxygen concentration followed by results of the simulations. It is found that oxygen extraction fraction, the fraction of oxygen removed from the blood in the capillary network by the tissue, and cerebral metabolic rate are dependent on both mean and heterogeneity of the transit time distribution. For changes in cerebral blood flow, a positive correlation can be observed between mean transit time and oxygen extraction fraction, and a negative correlation between mean transit time and metabolic rate of oxygen. A negative correlation can also be observed between transit time heterogeneity and the metabolic rate of oxygen for a constant cerebral blood flow. A sensitivity analysis on the mean and heterogeneity of the transit time distribution was able to quantify their respective contributions to oxygen extraction fraction and metabolic rate of oxygen. Mean transit time has a greater contribution than the heterogeneity for oxygen extraction fraction. This is found to be opposite for metabolic rate of oxygen. These results provide information on the role of the cerebral microvasculature and its effects on flow and metabolism. They thus open up the possibility of obtaining additional valuable clinical information for diagnosing and treating cerebrovascular diseases.

Multi-scale homogenization of blood flow in 3-dimensional human cerebral microvascular networks

The microvasculature plays a crucial role in the perfusion of blood through cerebral tissue. Current models of the cerebral microvasculature are discrete, and hence only able to model the perfusion over small voxel sizes before becoming computationally prohibitive. Larger models are required to provide comparisons and validation against imaging data. In this work, multi-scale homogenization methods were employed to develop continuum models of blood flow in a capillary network model of the human cortex. Homogenization of the local scale blood flow equations produced an averaged form of Darcy׳s law, with the permeability tensor encapsulating the capillary bed topology. A statistically accurate network model of the human cortex microvasculature was adapted to impose periodicity, and the elements of the permeability tensor calculated over a range of voxel sizes. The permeability tensor was found to converge to an effective permeability as voxel size increased. This converged permeability tensor was isotropic, reflecting the mesh-like structure of the cerebral microvasculature, with off-diagonal terms normally distributed about zero. A representative elementary volume of 375µm, with a standard deviation of 4.5% from the effective permeability, was determined. Using the converged permeability values, the cerebral blood flow was calculated to be around 55mLmin(-1)100g(-1), which is in very close agreement with experimental values. These results open up the possibility of future multi-scale modeling of the cerebral vascular network.

Comparison of generated parallel capillary arrays to three-dimensional reconstructed capillary networks in modeling oxygen transport in discrete microvascular volumes

OBJECTIVE

We compare RMN to PCA under several simulated physiological conditions to determine how the use of different vascular geometry affects oxygen transport solutions.

METHODS

Three discrete networks were reconstructed from intravital video microscopy of rat skeletal muscle (84 × 168 × 342 μm, 70 × 157 × 268 μm, and 65 × 240 × 571 μm), and hemodynamic measurements were made in individual capillaries. PCAs were created based on statistical measurements from RMNs. Blood flow and O₂ transport models were applied, and the resulting solutions for RMN and PCA models were compared under four conditions (rest, exercise, ischemia, and hypoxia).

RESULTS

Predicted tissue PO₂ was consistently lower in all RMN simulations compared to the paired PCA. PO₂ for 3D reconstructions at rest were 28.2 ± 4.8, 28.1 ± 3.5, and 33.0 ± 4.5 mmHg for networks I, II, and III compared to the PCA mean values of 31.2 ± 4.5, 30.6 ± 3.4, and 33.8 ± 4.6 mmHg. Simulated exercise yielded mean tissue PO₂ in the RMN of 10.1 ± 5.4, 12.6 ± 5.7, and 19.7 ± 5.7 mmHg compared to 15.3 ± 7.3, 18.8 ± 5.3, and 21.7 ± 6.0 in PCA.

CONCLUSIONS

These findings suggest that volume matched PCA yield different results compared to reconstructed microvascular geometries when applied to O₂ transport modeling; the predominant characteristic of this difference being an over estimate of mean tissue PO₂. Despite this limitation, PCA models remain important for theoretical studies as they produce PO₂ distributions with similar shape and parameter dependence as RMN.

Modelling of the physiological response of the brain to ischaemic stroke

Identification of salvageable brain tissue is a major challenge when planning the treatment of ischaemic stroke. As the standard technique used in this context, the perfusion-diffusion mismatch, has not shown total accuracy, there is an ongoing search for new imaging protocols that could better identify the region of the brain at risk and for new physiological models that could, on the one hand, incorporate the imaged parameters and predict the evolution of the condition for the individual, and, on the other hand, identify future biomarkers and thus suggest new directions for the design of imaging protocols. Recently, models of cellular metabolism after stroke and blood-brain barrier transport at tissue level have been introduced. We now extend these results by developing a model of the propagation of key metabolites in the brain's extracellular space owing to stroke-related oedema and chemical concentration gradients between the ischaemic and normal brain. We also couple the resulting chemical changes in the extracellular space with cellular metabolism. Our work enables the first patient-specific simulations of stroke progression with finite volume models to be made.

A generalized mathematical framework for estimating the residue function for arbitrary vascular networks

The microvasculature plays a vital part in the cardiovascular system. Any impairment to its function can lead to significant pathophysiological effects, particularly in organs such as the brain where there is a very tight coupling between structure and function. However, it is extremely difficult to quantify the health of the microvasculature in vivo, other than by assessing perfusion, using techniques such as arterial spin labelling. Recent work has suggested that the flow distribution within a voxel could also be a valuable measure. This can also be measured clinically, but as yet has not been related to the properties of the microvasculature due to the difficulties in modelling and characterizing these strongly inter-connected networks. In this paper, we present a new technique for characterizing an existing physiologically accurate model of the cerebral microvasculature in terms of its residue function. A new analytical mathematical framework for calculation of the residue function, based on the mass transport equation, of any arbitrary network is presented together with results from simulations. We then present a method for characterizing this function, which can be directly related to clinical data, and show how the resulting parameters are affected under conditions of both reduced perfusion and reduced network density. It is found that the residue function parameters are affected in different ways by these two effects, opening up the possibility of using such parameters, when acquired from clinical data, to infer information about both the network properties and the perfusion distribution. These results open up the possibility of obtaining valuable clinical information about the health of the microvasculature in vivo, providing additional tools to clinicians working in cerebrovascular diseases, such as stroke and dementia.

Cerebral microcirculation and oxygen tension in the human secondary cortex

The three-dimensional spatial arrangement of the cortical microcirculatory system is critical for understanding oxygen exchange between blood vessels and brain cells. A three-dimensional computer model of a 3 × 3 × 3 mm(3) subsection of the human secondary cortex was constructed to quantify oxygen advection in the microcirculation, tissue oxygen perfusion, and consumption in the human cortex. This computer model accounts for all arterial, capillary and venous blood vessels of the cerebral microvascular bed as well as brain tissue occupying the extravascular space. Microvessels were assembled with optimization algorithms emulating angiogenic growth; a realistic capillary bed was built with space filling procedures. The extravascular tissue was modeled as a porous medium supplied with oxygen by advection-diffusion to match normal metabolic oxygen demand. The resulting synthetic computer generated network matches prior measured morphometrics and fractal patterns of the cortical microvasculature. This morphologically accurate, physiologically consistent, multi-scale computer network of the cerebral microcirculation predicts the oxygen exchange of cortical blood vessels with the surrounding gray matter. Oxygen tension subject to blood pressure and flow conditions were computed and validated for the blood as well as brain tissue. Oxygen gradients along arterioles, capillaries and veins agreed with in vivo trends observed recently in imaging studies within experimental tolerances and uncertainty.

Multiscale imaging and computational modeling of blood flow in the tumor vasculature

The evolution in our understanding of tumor angiogenesis has been the result of pioneering imaging and computational modeling studies spanning the endothelial cell, microvasculature and tissue levels. Many of these primary data on the tumor vasculature are in the form of images from pre-clinical tumor models that provide a wealth of qualitative and quantitative information in many dimensions and across different spatial scales. However, until recently, the visualization of changes in the tumor vasculature across spatial scales remained a challenge due to a lack of techniques for integrating micro- and macroscopic imaging data. Furthermore, the paucity of three-dimensional (3-D) tumor vascular data in conjunction with the challenges in obtaining such data from patients presents a serious hurdle for the development and validation of predictive, multiscale computational models of tumor angiogenesis. In this review, we discuss the development of multiscale models of tumor angiogenesis, new imaging techniques capable of reproducing the 3-D tumor vascular architecture with high fidelity, and the emergence of "image-based models" of tumor blood flow and molecular transport. Collectively, these developments are helping us gain a fundamental understanding of the cellular and molecular regulation of tumor angiogenesis that will benefit the development of new cancer therapies. Eventually, we expect this exciting integration of multiscale imaging and mathematical modeling to have widespread application beyond the tumor vasculature to other diseases involving a pathological vasculature, such as stroke and spinal cord injury.