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Relative frequencies of (A) asymmetry ratio, (B) area ratio, (C) length‐to‐radius‐ratio, (D) tapering.
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Quantitative data on branching patterns of the human cerebral arterial tree are lacking in the 1.0–0.1 mm radius range. We aimed to collect quantitative data in this range, and to study if the cerebral artery tree complies with the principle of minimal work (Law of Murray).
To enable easy quantification of branching patterns a semi‐automatic method...
Citations
... Reliable measurements are challenging to obtain and are 37 therefore rarely employed [12]. Instead, the Principle of Minimum Work can be used 38 to prescribe the flow split among the distal ends of the simulated network based on the 39 relationship of their cross-sectional areas [13]. This heuristic, arising from the pursuit of 40 minimizing the energy spent on the transport and storage of blood, has been confirmed 41 through ex-vivo analysis by analyzing the regularity of arterial branching patterns [14]. ...
... Using an exponent of (n = 2) in Murray's law (see Equation (4)) reduces the discrepancy for the flow in the MCA, though at the cost of overestimating the flows in smaller vessels such as the OA. Detailed measurements of brain arterial trees [13] suggests that Murray's law is best fit by exponents that grow from 2 in the proximal sections towards 3 in the more distal sections. While this factor is important to consider in general, we opted for a consistent modelling of the distal systems based on the Poiseulle flow. ...
Hemodynamic simulations are increasingly used to study vascular diseases such as Intracranial Aneurysms (IA) and to further develop treatment options. However, due to limited data, certain aspects must rely on heuristics, especially at the simulation’s distal ends. In the literature, Murray’s Law is often used to model the outflow split based on vessel cross-section area; however, this poses challenges for the communicating arteries in the Circle of Willis (CoW). In this study, we contribute by assessing the impact of Murray’s Law in patient-specific geometries featuring IA at the posterior communication. We simulate different domain extensions representing common modelling choices and establish Full CoW simulations as a baseline to evaluate the effect of these modelling assumptions on hemodynamic indicators, focusing on IA growth and rupture-related factors such as the Wall Shear Stress (WSS) and Oscillatory Shear Index (OSI). Our findings reveal qualitative alterations in hemodynamics when not modeling posterior communication. Comparisons between computing the anterior circulation and computing the whole Circle of Willis reveal that quantitative changes in WSS may reach up to 80%, highlighting the significance of modelling choices in assessing IA risks and treatment strategies.
... Instead, the Principle of Minimum Work can be used to prescribe the flow split among the distal ends of the simulated network based on the relationship of their cross-section areas. This heuristic, which stems from the pursuit of minimizing the energy spent on the transport and storage of blood, has been confirmed through ex-vivo analysis by analyzing the regularity of the arterial branching patterns [11]. The Law of Minimum Work (also referred to as Murray's law), apart from being more realistic than plain stress-free outflows [12,13], bears the benefit with respect to the latter that it solely depends on geometrical features of the network. ...
Haemodynamic simulations are increasingly used to study vascular diseases like Intracranial Aneurysms (IA) and to further develope treatment options. However, due to limited data, some aspects must rely on heuristics, especially at the simulation’s distal ends. In literature, Murray’s Law is often used to model the outflow split based on vessel cross-section area, but this poses challenges for the communicating arteries in the Circle of Willis (CoW). In this study, we contribute by assessing the impact of Murray’s Law in patient-specific geometries featuring IA at the posterior communication. We simulate different domain extensions, representing common modelling choices. We establish Full CoW simulations as a baseline to evaluate the effect of these modelling assumptions on haemodynamic indicators, focusing on IA growth and rupture-related factors like Wall Shear Stress (WSS) and Oscillatory Shear Index (OSI). Our findings reveal qualitative alterations in haemodynamics when not modeling posterior communication. Comparisons between computing the anterior circulation and computing the whole Circle of Willis reveal quantitative changes in WSS may reach up to 80%, highlighting the significance of modelling choices in assessing IA risks and treatment strategies.
... Although time-offlight imaging using 7-Tesla magnetic resonance imaging can now non-invasively detect arterial structures with diameters of >100 μm [35], a relatively low flow rate causes low signal intensity and makes detection challenging compared with other structures of similar diameters. Furthermore, the numbers of intermediate anastomoses are few, and these structures were even not mentioned in a recent cadaver study of cerebral arterial networks with diameters of >100 μm [36]; these features point to difficulties in patient-specific morphological understanding and intermediate anastomose measurements. Further research into the patientspecific detection of arterial anastomoses is thus needed. ...
... Although this idealization is effective for expressing the global morphological characteristics shown in [10] with a simple algorithm, it does not reflect the hierarchical loop structures identified in the original study [10]. The branching patterns of large arteries, including cerebral arteries with a diameter of >O(10 2 ) μm [36], obey the minimum energy principle [24]; however, this principle does not accurately express the morphological characteristics of microvasculature (O(10 0 )-O(10 1 ) μm) in the human cerebral cortex [40]. The pial arterial network on the brain cortex is located between the aforementioned arterial structures as an interface; thus, further consideration of the functional importance of the pial arterial network will extend our conventional understanding of the physiological optimality of multiscale arterial structures. ...
The cerebral arterial network covering the brain cortex has multiscale anastomosis structures with sparse intermediate anastomoses (O[102] μm in diameter) and dense pial networks (O[101] μm in diameter). Recent studies indicate that collateral blood supply by cerebral arterial anastomoses has an essential role in the prognosis of acute ischemic stroke caused by large vessel occlusion. However, the physiological importance of these multiscale morphological properties-and especially of intermediate anastomoses-is poorly understood because of innate structural complexities. In this study, a computational model of multiscale anastomoses in whole-brain-scale cerebral arterial networks was developed and used to evaluate collateral blood supply by anastomoses during middle cerebral artery occlusion. Morphologically validated cerebral arterial networks were constructed by combining medical imaging data and mathematical modeling. Sparse intermediate anastomoses were assigned between adjacent main arterial branches; the pial arterial network was modeled as a dense network structure. Blood flow distributions in the arterial network during middle cerebral artery occlusion simulations were computed. Collateral blood supply by intermediate anastomoses increased sharply with increasing numbers of anastomoses and provided one-order-higher flow recoveries to the occluded region (15%-30%) compared with simulations using a pial network only, even with a small number of intermediate anastomoses (≤10). These findings demonstrate the importance of sparse intermediate anastomoses, which are generally considered redundant structures in cerebral infarction, and provide insights into the physiological significance of the multiscale properties of arterial anastomoses.
... The branching pattern of the cerebral arterial vessels is a complex field that still poses many unanswered questions [18]. We see the following reasons for quantifying the directional asymmetry: clinical and anatomical sciences find structural lateralization of the cerebral structure and function important [28], potential clinical scenarios that might benefit from the fast and automatic computation of those indices rely on qualitative and quantitative evaluation of collateral flow in cerebral vessels. ...
Understanding the 3D cerebral vascular network is one of the pressing issues impacting the diagnostics of various systemic disorders and is helpful in clinical therapeutic strategies. Unfortunately, the existing software in the radiological workstation does not meet the expectations of radiologists who require a computerized system for detailed, quantitative analysis of the human cerebrovascular system in 3D and a standardized geometric description of its components. In this study, we show a method that uses 3D image data from magnetic resonance imaging with contrast to create a geometrical reconstruction of the vessels and a parametric description of the reconstructed segments of the vessels. First, the method isolates the vascular system using controlled morphological growing and performs skeleton extraction and optimization. Then, around the optimized skeleton branches, it creates tubular objects optimized for quality and accuracy of matching with the originally isolated vascular data. Finally, it optimizes the joints on n-furcating vessel segments. As a result, the algorithm gives a complete description of shape, position in space, position relative to other segments, and other anatomical structures of each cerebrovascular system segment. Our method is highly customizable and in principle allows reconstructing vascular structures from any 2D or 3D data. The algorithm solves shortcomings of currently available methods including failures to reconstruct the vessel mesh in the proximity of junctions and is free of mesh collisions in high curvature vessels. It also introduces a number of optimizations in the vessel skeletonization leading to a more smooth and more accurate model of the vessel network. We have tested the method on 20 datasets from the public magnetic resonance angiography image database and show that the method allows for repeatable and robust segmentation of the vessel network and allows to compute vascular lateralization indices.
Graphical abstract
... This method was used in the studies of the retinal vessels [24], the kidneys arterial tree [25], vessels of lungs [26], heart [27] and pituitary gland [28]. The vascular network of the brain also has fractal properties [29]. A quantitative assessment of the superficial vascular network of the cerebellum was performed using fractal analysis of anatomical preparations [30]. ...
Morphometry is an integral part of most modern morphological studies and the classic morphological morphometric methods and techniques are often borrowed for research in other fields of medicine. The majority of morphometric techniques are derived from Euclidean geometry. In the past decades, the principles, parameters and methods of fractal geometry are increasingly used in morphological studies. The basic parameter of fractal geometry is fractal dimension. Fractal dimension allows you to quantify the degree of filling of space with a certain geometric object and to characterize the complexity of its spatial configuration. There are many anatomical structures with complex irregular shapes that cannot be unambiguously and comprehensively characterized by methods and techniques of traditional geometry and traditional morphometry: irregular linear structures, irregular surfaces of various structures and pathological foci, structures with complex branched, tree-like, reticulated, cellular or porous structure, etc. Fractal dimension is a useful and informative morphometric parameter that can complement existing quantitative parameters to quantify objective characteristics of various anatomical structures and pathological foci. Fractal analysis can qualitatively complement existing morphometric methods and techniques and allow a comprehensive assessment of the spatial configuration complexity degree of irregular anatomical structures. The review describes the basic principles of Euclidean and fractal geometry and their application in morphology and medicine, importance and application of sizes and their derivatives, topological, metric and fractal dimensions, regular and irregular figures in morphology, and practical application of fractal dimension and fractal analysis in the morphological studies and clinical practice.
... Although extensive anatomical studies have described topological properties and the relevant constituents of the pial arterial vasculature (Pfeifer, 1930;Szikla et al., 1977), quantitative data of the human pial arterial vasculature remain scarce (Cassot et al., 2006;Helthuis et al., 2019;Hirsch et al., 2012;Payne, 2017;Schmid et al., 2019). The by far still 'most comprehensive and influential work' (Hirsch et al., 2012) is the detailed description of the pial vasculature by Duvernoy et al. (1981), which examined 25 brains using intravascular ink injections. ...
... As indispensable as this dataset has been, 3D reconstructions of the vascular network and surrounding anatomy were not provided in this study. A second recent analysis performed by Helthuis et al. (2019) used corrosion casts from four brain specimens and provided valuable insights into the branching pattern of the arterial vasculature. ...
... When applying measures of arterial topology and morphometry to address questions about the robustness of blood supply (Baumbach and Heistad, 1985), arterial territories (Mut et al., 2014) or the relationship between the cortical folding pattern and the pial vasculature (Ii et al., 2020), particular care needs to be exercised to account for the 'voxel size' bias. By this we mean that the dependency of the vessel contrast on the voxel size shown in our study in conjunction with the reduction in vessel diameter along the vascular tree (Duvernoy et al., 1981;Helthuis et al., 2019) can introduce a systematic detection bias that varies regionally. At lower imaging resolutions, we would expect particularly high numbers of false negatives for example in anterior and posterior brain regions, which are the end points of the pial vascular tree and whose arteries generally have lower vessel diameters. ...
The pial arterial vasculature of the human brain is the only blood supply to the neocortex, but quantitative data on the morphology and topology of these mesoscopic arteries (diameter 50-300µm) remains scarce. Because it is commonly assumed that blood flow velocities in these vessels are prohibitively slow, non-invasive time-of-flight MRI angiography (TOF-MRA)-which is well-suited to high 3D imaging resolutions-has not been applied to imaging the pial arteries. Here, we provide a theoretical framework that outlines how TOF-MRA can visualize small pial arteries in vivo, by employing extremely small voxels at the size of individual vessels. We then provide evidence for this theory by imaging the pial arteries at 140-µm isotropic resolution using a 7T MRI scanner and prospective motion correction, and show that pial arteries one voxel-width in diameter can be detected. We conclude that imaging pial arteries is not limited by slow blood flow, but instead by achievable image resolution. This study represents the first targeted, comprehensive account of imaging pial arteries in vivo in the human brain. This ultra-high-resolution angiography will enable the characterization of pial vascular anatomy across the brain to investigate patterns of blood supply and relationships between vascular and functional architecture.
... Beyond these general properties, quantitative data on the topology of the human pial arterial vasculature remain scarce (Cassot et al., 2006;Helthuis et al., 2019;Hirsch et al., 2012;Payne, 2017;Schmid et al., 2019). The by far still 'most comprehensive and influential work' (Hirsch et al., 2012) is the detailed description of the pial vasculature by Duvernoy et al. (1981), which examined 25 brains using intravascular ink injections. ...
... As indispensable as this dataset has been, 3D reconstructions of the vascular network and surrounding anatomy were not provided in this study. A second recent analysis performed by Helthuis et al. (2019) used corrosion casts from four brain specimens and provided valuable insights into the branching pattern of the arterial vasculature. However, only limited information can be obtained in this way about the morphometry of vessels, in particular their position and geometric relationship with the cortex. ...
... 1101 the relationship between the cortical folding pattern and the pial vasculature (Ii et al., 2020), particular care needs to be exercised to account for the 'voxel size' bias. By this we mean that the dependency of the vessel contrast on the voxel size shown in our study in conjunction with the reduction in vessel diameter along the vascular tree (Duvernoy et al., 1981;Helthuis et al., 2019), can introduce a systematic detection bias that varies regionally. At lower imaging resolutions, we would expect particularly high numbers of false negatives for example in anterior and posterior brain regions, which are the end points of the pial vascular tree and whose arteries generally have lower vessel diameters. ...
The pial arterial vasculature of the human brain is the only blood supply to the neocortex, but quantitative data on the morphology and topology of these mesoscopic vessels (diameter 50-300 μm) remains scarce. Because it is commonly assumed that blood flow velocities in these vessels are prohibitively slow, non-invasive time-of-flight MRI angiography (TOF-MRA)-which is well-suited to high 3D imaging resolutions-has not been applied to imaging the pial arteries. Here, we provide a theoretical framework that outlines how TOF-MRA can visualize small pial arteries in vivo, by employing extremely small voxels at the size of individual vessels. We then provide evidence for this theory by imaging the pial arteries at 140-μm isotropic resolution using a 7T MRI scanner and prospective motion correction, and show that pial arteries one voxel-width in diameter can be detected. We conclude that imaging pial arteries is not limited by slow blood flow, but instead by achievable image resolution. This study represents the first targeted, comprehensive account of imaging pial arteries in vivo in the human brain. This ultra-high-resolution angiography will enable the characterization of pial vascular anatomy across the brain to investigate patterns of blood supply and relationships between vascular and functional architecture.
... It would be interesting to see the effect of different scaling laws. Murray's Law is royalsocietypublishing.org/journal/rsfs Interface Focus 11: 20190125 used in this paper as it seems to be a good fit for the cerebral vasculature [48]. A simple alternative to the coupling algorithm would be to map the surface to the nearest outlet. ...
An acute ischaemic stroke is due to the sudden blockage of an intracranial blood vessel by an embolized thrombus. In the context of setting up in silico trials for the treatment of acute ischaemic stroke, the effect of a stroke on perfusion and metabolism of brain tissue should be modelled to predict final infarcted brain tissue. This requires coupling of blood flow and tissue perfusion models. A one-dimensional intracranial blood flow model and a method to couple this to a brain tissue perfusion model for patient-specific simulations is presented. Image-based patient-specific data on the anatomy of the circle of Willis are combined with literature data and models for vessel anatomy not visible in the images, to create an extended model for each patient from the larger vessels down to the pial surface. The coupling between arterial blood flow and tissue perfusion occurs at the pial surface through the estimation of perfusion territories. The coupling method is able to accurately estimate perfusion territories. Finally, we argue that blood flow can be approximated as steady-state flow at the interface between arterial blood flow and tissue perfusion to reduce the cost of organ-scale simulations.
... 10: 190249 ordering method was preferred. We subscribed to the consideration of Weibel [95] and Hsia et al. [129] that for the human lung, the morphogenetic ordering method provides more instructive data for understanding physiological processes such as flow dynamics [95,138,140,141] and particle deposition [142,143], while Strahler's ordering method yields more meaningful data for the pathologists [129,138]. Furthermore, Horsfield [138] cautioned that a great deal of information is lost in the simplification inherent in Strahler's ordering method, especially with regard to the connectivity of the branches. ...
... In biological structures, there is lack of unanimity on what constitutes optimization [150,151,165,171,172,190,191] and whether the state/condition is achievable or even desirable [121,[173][174][175][176][177]. Regarding the H-ML, some of the views of concern that have been expressed are the following: 'perhaps Murray's law should be viewed as more of what you would call "guidelines" than actual rules' [20]; 'optimum models are abstractions of biological systems and they are not expected to fit these systems with absolute accuracy' [140]; and 'there is a large spread between different parts of the circulation and possibly between different subjects in regard to the principal of minimum work' [141]. In complete departure from the orthodox thinking that optimization is an adaptive (i.e. ...
Fractal geometry (FG) is a branch of mathematics that instructively characterizes structural complexity. Branched structures are ubiquitous in both the physical and the biological realms. Fractility has therefore been termed nature's design. The fractal properties of the bronchial (airway) system, the pulmonary artery and the pulmonary vein of the human lung generates large respiratory surface area that is crammed in the lung. Also, it permits the inhaled air to intimately approximate the pulmonary capillary blood across a very thin blood-gas barrier through which gas exchange to occur by diffusion. Here, the bronchial (airway) and vascular systems were simultaneously cast with latex rubber. After corrosion, the bronchial and vascular system casts were physically separated and cleared to expose the branches. The morphogenetic (Weibel's) ordering method was used to categorize the branches on which the diameters and the lengths, as well as the angles of bifurcation, were measured. The fractal dimensions (DF) were determined by plotting the total branch measurements against the mean branch diameters on double logarithmic coordinates (axes). The diameter-determined DF values were 2.714 for the bronchial system, 2.882 for the pulmonary artery and 2.334 for the pulmonary vein while the respective values from lengths were 3.098, 3.916 and 4.041. The diameters yielded DF values that were consistent with the properties of fractal structures (i.e. self-similarity and space-filling). The data obtained here compellingly suggest that the design of the bronchial system, the pulmonary artery and the pulmonary vein of the human lung functionally comply with the Hess-Murray law or 'the principle of minimum work'.
... As cerebral arteries are known to have minimal to no tapering, each artery was segmented as one segment (e.g. middle cerebral artery M1 part would be one segment) (Helthuis et al., 2018b). Depending on the quality of the MRI, centrelines were drawn in arterial segments for the anterior, middle and posterior cerebral arteries up to the A3 (third anterior cerebral arterial branching level), M5 (fifth middle cerebral arterial branching level) and P3 (third posterior cerebral arterial branching level). ...
... Based on data by Helthuis et al. (2018b) AR was chosen to be 1.33 and c as 0.65. Finally, the scaling factors were calculated as described by Olufsen (1999) according to ...
... Arterial blood pressure was used as a patientspecific proximal boundary condition. To enable more easy-to-use patient-specific distal boundary conditions, a structured tree was used based on the particular branching patterns as measured previously (Helthuis et al., 2018b). ...
In clinical practice, many complex choices in treatment of complex cerebrovascular diseases have to be made. A patient-specific mathematical blood flow could aid these decisions. For certain cases, less accuracy is required and more simplistic models might be feasible. The current study is aiming to validate a patient-specific simplistic blood flow model in 20 healthy subjects. All subjects underwent MRI and Noninvasive Optimal Vessel Analysis (NOVA) to obtain patient-specific vascular morphology and flow measurements of all major cerebral arteries for validation. The mathematical model used was based on the Hagen-Poiseuille equations. Proximal boundary conditions were patient-specific blood pressure cuff measurements. For distal boundary conditions, a structured tree and a simple autoregulatory model were applied. Autoregulatory parameters were optimized based on the data of 10 additional healthy subjects. A median percentual flow difference of -3% (interquartile range -36% to 17%) was found. Regression analysis to an identity line resulted in R2 values of 0.71 for absolute flow values. Bland-Altman plots showed a bias (levels of agreement) of 5% (-70 to 80%) for absolute flow. Based on these results the model proved to be accurate within a range that might be feasible for use in clinic. Major limitations to the model arise from the simplifications made compared to the actual physiological situation and limitations in the validation method. As the model is validated in healthy subjects only, further validation in actual patients is needed.
