A systematic review and meta-analysis of Murray's Law in the Coronary Arterial Circulation

Murray's law has been viewed as a fundamental law of physiology. Relating blood flow (Q) to vessel diameter (D) (Q µD3), it dictates minimum lumen area (MLA) targets for coronary bifurcation percutaneous coronary intervention (PCI). The cubic exponent (3.0) however, has long been disputed, with alternative theoretical derivations arguing this should be closer to 2.33 (7/3). The aim of this meta-analysis was to quantify the optimum flow-diameter exponent in human and mammalian coronary arteries. We conducted a systematic review and meta-analysis of all articles quantifying an optimum flow-diameter exponent for mammalian coronary arteries within the Cochrane library, PubMed Medline, Scopus, and Embase databases on 20/3/2023. A random-effects meta-analysis was used to determine a pooled flow-diameter exponent. Risk of bias was assessed with the National Institutes of Health (NIH) quality assessment tool, funnel plots and Egger regression. From a total of 4,524 articles, 18 were suitable for meta-analysis. Studies included data from 1,070 unique coronary trees, taken from 372 humans and 112 animals. The pooled flow diameter exponent across both epicardial and transmural arteries was 2.39 (95% CI 2.24 - 2.54, I2 = 99%). The pooled exponent of 2.39 showed very close agreement with the theoretical exponent of 2.33 (7/3) reported by Kassab and colleagues. This exponent may provide a more accurate description of coronary morphometric scaling in human and mammalian coronary arteries, as compared with Murray's original law. This has important implications for the assessment, diagnosis, and interventional treatment of coronary artery disease.

In silico analysis of a hierarchical microfluidic vascular network: Detecting the location of angiogenic sprouting

Lack of oxygen is one of the leading causes of failure in engineered tissue. Therefore, angiogenesis will be necessary for the survival of larger tissues in vivo. In addition, a proper lymphatic system that plays an essential role in relieving inflammation and maintaining tissue homeostasis is of great importance for tissue regeneration and repair. Many biomechanical parameters are involved in controlling angiogenesis and capillary network generation, which are challenging to study and control in experimental studies or in vitro. In the present study, using numerical modeling, the effect of various geometric and biomechanical parameters in creating suitable conditions for angiogenesis was investigated. Furthermore, sprouting points were predicted using flow dynamics. For this purpose, a porous scaffold, flow channels with parametric geometry that followed Murray's law, and a drainage channel were considered. Results suggested that the geometry of the microfluidic channels and the characteristics of the vessel wall and scaffold plays a complementary role in determining the transmural pressure. It was found that a twofold increase in the vascular hydraulic conductivity can reduce the minimum transmural pressure by up to 28% and increase the drainage flow rate by 44%. In addition, the minimum magnitude of transmural pressure tends to zero for scaffold's hydraulic conductivity values smaller than 10^-11  m³ s kg^-1 . The results of this study can be used in optimizing the design of the relevant microfluidic systems to induce angiogenesis and avoid leakage in the constructed implantable tissue.

An optimal control approach to determine resistance-type boundary conditions from in-vivo data for cardiovascular simulations

The choice of appropriate boundary conditions is a fundamental step in computational fluid dynamics (CFD) simulations of the cardiovascular system. Boundary conditions, in fact, highly affect the computed pressure and flow rates, and consequently haemodynamic indicators such as wall shear stress (WSS), which are of clinical interest. Devising automated procedures for the selection of boundary conditions is vital to achieve repeatable simulations. However, the most common techniques do not automatically assimilate patient-specific data, relying instead on expensive and time-consuming manual tuning procedures. In this work, we propose a technique for the automated estimation of outlet boundary conditions based on optimal control. The values of resistive boundary conditions are set as control variables and optimized to match available patient-specific data. Experimental results on four aortic arches demonstrate that the proposed framework can assimilate 4D-Flow MRI data more accurately than two other common techniques based on Murray's law and Ohm's law.

Image-guided simulation in comparison with laser speckle contrast imaging for full-field observation of blood flow in a microvasculature model

The in vitro reconstruction of the microvascular network model provides a reproducible platform for hemodynamic study with great biological relevance. In the present study, microvascular models with different parametric features were designed under the guidance of Murray's law and derived from representative natural vascular network topography in vivo. Computational fluid dynamics (CFD) was used to numerically simulate blood velocity distributions inside of the designed microvasculature models. Full-field blood flow in the vascular network was visualized in vivo using a laser speckle contrast imaging (LSCI) system, from which the measured relative velocity was compared with CFD computed flow distribution. The results have shown that, in comparison with the simplified flow patterns obtained from idealized geometries, the irregular vascular topography is expected to lead to nonuniform and poor regional blood velocity distribution. The velocity distribution acquired by in vivo LSCI experiment is in good agreement with that of numerical simulation, indicating the technical feasibility of using biomimetic microchannels as a reasonable approximation of the microcirculatory flow conditions. This study provides a new paradigm that can be well suited to the study of microvascular blood flow properties and can further expand to mimic other in-vivo scenarios for accurately recapitulating the physical and hemodynamic environment of the microcirculation.

Interspecific scaling of blood flow rates and arterial sizes in mammals

This meta-study investigated the relationships between blood flow rate (Q̇; cm³ s^-1), wall shear stress (τ_w; dyn cm^-2) and lumen radius (r _i; cm) in 20 named systemic arteries of nine species of mammals, ranging in mass from 23 g mice to 652 kg cows, at rest. In the dataset, derived from 50 studies, lumen radius varied between 3.7 µm in a cremaster artery of a rat and 11.2 mm in the aorta of a human. The 92 logged data points of [Formula: see text] and r _i are described by a single second-order polynomial curve with the equation: [Formula: see text] The slope of the curve increased from approximately 2 in the largest arteries to approximately 3 in the smallest ones. Thus, da Vinci's rule ([Formula: see text]) applies to the main arteries and Murray's law ([Formula: see text]) applies to the microcirculation. A subset of the data, comprising only cephalic arteries in which [Formula: see text] is fairly constant, yielded the allometric power equation: [Formula: see text] These empirical equations allow calculation of resting perfusion rates from arterial lumen size alone, without reliance on theoretical models or assumptions on the scaling of wall shear stress in relation to body mass. As expected, [Formula: see text] of individual named arteries is strongly affected by body mass; however, [Formula: see text] of the common carotid artery from six species (mouse to horse) is also sensitive to differences in whole-body basal metabolic rate, independent of the effect of body mass.

Vessel scaling in evergreen angiosperm leaves conforms with Murray's law and area-filling assumptions: implications for plant size, leaf size and cold tolerance

Water transport in leaf vasculature is a fundamental process affecting plant growth, ecological interactions and ecosystem productivity, yet the architecture of leaf vascular networks is poorly understood. Although Murray's law and the West-Brown-Enquist (WBE) theories predict convergent scaling of conduit width and number, it is not known how conduit scaling is affected by habitat aridity or temperature. We measured the scaling of leaf size, conduit width and conduit number within the leaves of 36 evergreen Angiosperms spanning a large range in aridity and temperature in eastern Australia. Scaling of conduit width and number in midribs and 2° veins did not differ across species and habitats (P > 0.786), and did not differ from that predicted by Murray's law (P = 0.151). Leaf size was strongly correlated with the hydraulic radius of petiole conduits (r²  = 0.83, P < 0.001) and did not differ among habitats (P > 0.064), nor did the scaling exponent differ significantly from that predicted by hydraulic theory (P = 0.086). The maximum radius of conduits in petioles was positively correlated with the temperature of the coldest quarter (r²  = 0.67; P < 0.001), suggesting that habitat temperature restricts the occurrence of wide-conduit species in cold habitats.

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.

The hydraulic architecture of Ginkgo leaves

PREMISE OF THE STUDY

The hydraulics of xylem has been widely studied in numerous species and organ types. However, comparatively little is known about how phloem and xylem are hydraulically coupled or about many of the basic structural properties of phloem (such as conducting cell numbers and conductive areas), which nevertheless have direct bearing on understanding phloem loading and unloading.

METHODS

Using a combination of light, epifluorescence, confocal, and transmission electron microscopy, we quantified the hydraulic architecture of Ginkgo biloba leaf laminae and examined the scaling relationships between phloem and xylem in five fully mature leaves.

KEY RESULTS

The conductive areas and lengths of sieve cells and tracheids increase basipetally toward the petiole in a manner that is consistent with Münch's pressure flow hypothesis for phloem transport. This trend holds true for individual veins, the sum of conductive areas across all veins at any distance from the petiole, and for individual sieve cells and tracheids. Further, the conductive areas of phloem and xylem are isometrically correlated across the entire vasculature of the leaf lamina. The data for conducting cell areas do not conform with the predictions of the hydraulic models of da Vinci and Murray.

CONCLUSIONS

The scaling of Ginkgo lamina hydraulics complies with that observed in leaves of other gymnosperms and most angiosperms and is inconsistent with theoretical models that assume that the volume of transported incompressible fluids is conserved.

Fundamental principles of vascular network topology

The vascular system is arguably the most important biological system in many organisms. Although the general principles of its architecture are simple, the growth of blood vessels occurs under extreme physical conditions. Optimization is an important aspect of the development of computational models of the vascular branching structures. This review surveys the approaches used to optimize the topology and estimate different geometrical parameters of the vascular system. The review is focused on optimizations using complex cost functions based on the minimum total energy principle and the relationship between the laws of growth and precise vascular network topology. Experimental studies of vascular networks in different species are also discussed.

A generalized optimization principle for asymmetric branching in fluidic networks

When applied to a branching network, Murray’s law states that the optimal branching of vascular networks is achieved when the cube of the parent channel radius is equal to the sum of the cubes of the daughter channel radii. It is considered integral to understanding biological networks and for the biomimetic design of artificial fluidic systems. However, despite its ubiquity, we demonstrate that Murray’s law is only optimal (i.e. maximizes flow conductance per unit volume) for symmetric branching, where the local optimization of each individual channel corresponds to the global optimum of the network as a whole. In this paper, we present a generalized law that is valid for asymmetric branching, for any cross-sectional shape, and for a range of fluidic models. We verify our analytical solutions with the numerical optimization of a bifurcating fluidic network for the examples of laminar, turbulent and non-Newtonian fluid flows.

Structure and hemodynamics of vascular networks in the chorioallantoic membrane of the chicken

The chick chorioallantoic membrane (CAM) is extensively used as an in vivo model. Here, structure and hemodynamics of CAM vessel trees were analyzed and compared with predictions of Murray's law. CAM microvascular networks of Hamburger-Hamilton stage 40 chick embryos were scanned by videomicroscopy. Three networks with ∼3,800, 580, and 480 segments were digitally reconstructed, neglecting the capillary mesh. Vessel diameters (D) and segment lengths were measured, and generation numbers and junctional exponents at bifurcations were derived. In selected vessels, flow velocities (v) and hematocrit were measured. Hemodynamic simulations, incorporating the branching of capillaries from preterminal vessels, were used to estimate v, volume flow, shear stress (τ), and pressure for all segments of the largest network. For individual arteriovenous flow pathways, terminal arterial and venous generation numbers are negatively correlated, leading to low variability of total topological and morphological pathway lengths. Arteriolar velocity is proportional to diameter (v∝D^1.03 measured, v∝D^0.93 modeling), giving nearly uniform τ levels (τ∝D^0.05). Venular trees exhibit slightly higher exponents (v∝D^1.3, τ∝D^0.38). Junctional exponents at divergent and convergent bifurcations were 2.05 ± 1.13 and 1.97 ± 0.95 (mean ± SD) in contrast to the value 3 predicted by Murray's law. In accordance with Murray's law, τ levels are (nearly) maintained in CAM arterial (venular) trees, suggesting vascular adaptation to shear stress. Arterial and venous trees show an interdigitating arrangement providing homogeneous flow pathway properties and have preterminal capillary branches. These properties may facilitate efficient oxygen exchange in the CAM during rapid embryonic growth.

Hematocrit dispersion in asymmetrically bifurcating vascular networks

Quantitative modeling of physiological processes in vasculatures requires an accurate representation of network topology, including vessel branching. We propose a new approach for reconstruction of vascular network, which determines how vessel bifurcations distribute red blood cells (RBC) in the microcirculation. Our method follows the foundational premise of Murray's law in postulating the existence of functional optimality of such networks. It accounts for the non-Newtonian behavior of blood by allowing the apparent blood viscosity to vary with discharge hematocrit and vessel radius. The optimality criterion adopted in our approach is the physiological cost of supplying oxygen to the tissue surrounding a blood vessel. Bifurcation asymmetry is expressed in terms of the amount of oxygen consumption associated with the respective tissue volumes being supplied by each daughter vessel. The vascular networks constructed with our approach capture a number of physiological characteristics observed in in vivo studies. These include the nonuniformity of wall shear stress in the microcirculation, the significant increase in pressure gradients in the terminal sections of the network, the nonuniformity of both the hematocrit partitioning at vessel bifurcations and hematocrit across the capillary bed, and the linear relationship between the RBC flux fraction and the blood flow fraction at bifurcations.

Radius exponent in elastic and rigid arterial models optimized by the least energy principle

It was analyzed in normal physiological arteries whether the least energy principle would suffice to account for the radius exponent x. The mammalian arterial system was modeled as two types, the elastic or the rigid, to which Bernoulli's and Hagen-Poiseuille's equations were applied, respectively. We minimized the total energy function E, which was defined as the sum of kinetic, pressure, metabolic and thermal energies, and loss of each per unit time in a single artery transporting viscous incompressible blood. Assuming a scaling exponent α between the vessel radius (r) and length (l) to be 1.0, x resulted in 2.33 in the elastic model. The rigid model provided a continuously changing x from 2.33 to 3.0, which corresponded to Uylings' and Murray's theories, respectively, through a function combining Reynolds number with a proportional coefficient of the l - r relationship. These results were expanded to an asymmetric arterial fractal tree with the blood flow preservation rule. While x in the optimal elastic model accounted for around 2.3 in proximal systemic (r >1 mm) and whole pulmonary arteries (r ≥0.004 mm), optimal x in the rigid model explained 2.7 in elastic-muscular (0.1 < r ≤1 mm) and 3.0 in peripheral resistive systemic arteries (0.004 ≤ r ≤0.1 mm), in agreement with data obtained from angiographic, cast-morphometric, and in vivo experimental studies in the literature. The least energy principle on the total energy basis provides an alternate concept of optimality relating to mammalian arterial fractal dimensions under α = 1.0.