Extension of Murray's law using a non-Newtonian model of blood flow

Non-invasive volumetric ultrasound localization microscopy detects vascular changes in mice with Alzheimer's disease

Alzheimer's Disease (AD) is the most common form of dementia and one of the leading causes of death. AD is known to be correlated to tortuosity in the microvasculature as well as decreases in blood flow throughout the brain. However, the mechanisms behind these changes and their causal relation to AD are poorly understood. Methods: Here, we use volumetric ultrasound localization microscopy (ULM) to non-invasively and quantitatively compare the microvascular morphology and flow dynamics of five wildtype (WT) and five APPNL-G-F Knock-in mice, a mouse model of AD, across a 1cmx1cmx1cm brain volume and in four specific brain regions: the hippocampal formation, thalamus, hypothalamus, and cerebral cortex. Results: Comparisons between groups showed a significant increase in tortuosity, as measured by the Sum of Angles Metric (SOAM), throughout the brain (p < 0.01) and the hypothalamus (p = 0.01), in mice with AD. While differences in mean velocity (p < 0.01) and blood flow (p=0.04) were detected across the whole brain, their effect size was small and no differences were detected in the four selected regions. There was a significant decrease in the linear log relationship between vessel diameter and blood flow, with AD mice experiencing a lower slope than WT mice across the whole brain volume (p = 0.02) and in the hippocampal formation (p = 0.05), a region affected by Amyloid Beta plaques in this mouse model. The AD mice had higher blood flows in smaller vessels and smaller blood flows in larger vessels than the WT mice. Conclusions: This preliminary demonstrates that the imaging technique can be used for non-invasive, longitudinal, volumetric assessment of AD, which may allow for investigation into the poorly understood microvascular degeneration associated with AD through time as well as the development of early diagnostic techniques.

Predicting straw drinking ability of liquid foods by pipe-flow rheometry

Designing the straw drinking experience is increasingly important in consumer-oriented liquid food innovation. The perceived 'straw drinking ease' of liquid foods could be assessed by sensory analysis. However, challenges arise in linking instrumental measurements to the subjective quantification of sensory attributes due to the complexity of perceptual behaviors in drinking. Here, we developed a pipe-flow rheometry approach to predict straw drinking ability with minimized reliance on panel tests. We measured the instantaneous flow of liquid samples within straws and compared their flow profiles based on different yielding behaviors. The dynamic flow processes were simplified into steady-state pipe flow, and perceived straw drinking ability was modeled as the shear viscosity of liquid foods at specific flow rates. A power-law relationship was found between viscosity at sample-specific shear rates and straw drinking ability, regardless of whether the liquid foods exhibit yield stress. This approach provides opportunities for directly addressing the straw drinking experience through simple laboratory measurements when developing texture modulation systems for liquid foods.

"Filler-Associated Acute Stroke Syndrome": Classification, Predictive Modelling of Hyaluronidase Efficacy, and Updated Case Review on Neurological and Visual Complications

INTRODUCTION

The rising use of soft tissue fillers for aesthetic procedures has seen an increase in complications, including vascular occlusions and neurological symptoms that resemble stroke. This study synthesizes information on central nervous system (CNS) complications post-filler injections and evaluates the effectiveness of hyaluronidase (HYAL) treatment.

METHODS

A thorough search of multiple databases, including PubMed, EMBASE, Scopus, Web of Science, Google Scholar, and Cochrane, focused on publications from January 2014 to January 2024. Criteria for inclusion covered reviews and case reports that documented CNS complications related to soft tissue fillers. Advanced statistical and computational techniques, including logistic regression, machine learning, and Bayesian analysis, were utilized to dissect the factors influencing therapeutic outcomes.

RESULTS

The analysis integrated findings from 20 reviews and systematic analyses, with 379 cases reported since 2018. Hyaluronic acid (HA) was the most commonly used filler, particularly in nasal region injections. The average age of patients was 38, with a notable increase in case reports in 2020. Initial presentation data revealed that 60.9% of patients experienced no light perception, while ptosis and ophthalmoplegia were present in 54.3 and 42.7% of cases, respectively. The statistical and machine learning analyses did not establish a significant linkage between the HYAL dosage and patient recovery; however, the injection site emerged as a critical determinant.

CONCLUSION

The study concludes that HYAL treatment, while vital for managing complications, varies in effectiveness based on the injection site and the timing of administration. The non-Newtonian characteristics of HA fillers may also affect the incidence of complications. The findings advocate for tailored treatment strategies incorporating individual patient variables, emphasizing prompt and precise intervention to mitigate the adverse effects of soft tissue fillers.

LEVEL OF EVIDENCE III

This journal requires that authors assign a level of evidence to each article. For a full description of these Evidence-Based Medicine ratings, please refer to the Table of Contents or the online Instructions to Authors www.springer.com/00266 .

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 ([Formula: see text]) to vessel diameter (D) ([Formula: see text]·∝·D3), 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 March 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 4,772 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% confidence interval: 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.

Analyzing solitary wave solutions of the nonlinear Murray equation for blood flow in vessels with non-uniform wall properties

Solitary wave solutions are of great interest to bio-mathematicians and other scientists because they provide a basic description of nonlinear phenomena with many practical applications. They provide a strong foundation for the development of novel biological and medical models and therapies because of their remarkable behavior and persistence. They have the potential to improve our comprehension of intricate biological systems and help us create novel therapeutic approaches, which is something that researchers are actively investigating. In this study, solitary wave solutions of the nonlinear Murray equation will be discovered using a modified extended direct algebraic method. These solutions represent a uniform variation in blood vessel shape and diameter that can be used to stimulate blood flow in patients with cardiovascular disease. These solutions are newly in the literature, and give researchers an important tool for grasping complex biological systems. To see how the solitary wave solutions behave, graphs are displayed using Matlab.

Integrative modeling of hemodynamic changes and perfusion impairment in coronary microvascular disease

Introduction: Coronary microvascular disease is one of the responsible factors for cardiac perfusion impairment. Due to diagnostic and treatment challenges, this pathology (characterized by alterations to microvasculature local hemodynamics) represents a significant yet unsolved clinical problem. Methods: Due to the poor understanding of the onset and progression of this disease, we propose a new and noninvasive strategy to quantify in-vivo hemodynamic changes occurring in the microvasculature. Specifically, we here present a conceptual workflow that combines both in-vitro and in-silico modelling for the analysis of the hemodynamic alterations in the microvasculature. Results: First, we demonstrate a hybrid additive manufacturing process to fabricate circular cross-section, biocompatible fluidic networks in polytetrafluoroethylene. We then use these microfluidic devices and computational fluid dynamics to simulate different degrees of perfusion impairment. Discussion: Ultimately, we show that the developed workflow defines a robust platform for the multiscale analysis of multifactorial events occurring in coronary microvascular disease.

Non-invasive estimation of the parameters of a three-element windkessel model of aortic arch arteries in patients undergoing thoracic endovascular aortic repair

Background: Image-based computational hemodynamic modeling and simulations are important for personalized diagnosis and treatment of cardiovascular diseases. However, the required patient-specific boundary conditions are often not available and need to be estimated. Methods: We propose a pipeline for estimating the parameters of the popular three-element Windkessel (WK3) models (a proximal resistor in series with a parallel combination of a distal resistor and a capacitor) of the aortic arch arteries in patients receiving thoracic endovascular aortic repair of aneurysms. Pre-operative and post-operative 1-week duplex ultrasound scans were performed to obtain blood flow rates, and intra-operative pressure measurements were also performed invasively using a pressure transducer pre- and post-stent graft deployment in arch arteries. The patient-specific WK3 model parameters were derived from the flow rate and pressure waveforms using an optimization algorithm reducing the error between simulated and measured pressure data. The resistors were normalized by total resistance, and the capacitor was normalized by total resistance and heart rate. The normalized WK3 parameters can be combined with readily available vessel diameter, brachial blood pressure, and heart rate data to estimate WK3 parameters of other patients non-invasively. Results: Ten patients were studied. The medians (interquartile range) of the normalized proximal resistor, distal resistor, and capacitor parameters are 0.10 (0.07-0.15), 0.90 (0.84-0.93), and 0.46 (0.33-0.58), respectively, for common carotid artery; 0.03 (0.02-0.04), 0.97 (0.96-0.98), and 1.91 (1.63-2.26) for subclavian artery; 0.18 (0.08-0.41), 0.82 (0.59-0.92), and 0.47 (0.32-0.85) for vertebral artery. The estimated pressure showed fairly high tolerance to patient-specific inlet flow rate waveforms using the WK3 parameters estimated from the medians of the normalized parameters. Conclusion: When patient-specific outflow boundary conditions are not available, our proposed pipeline can be used to estimate the WK3 parameters of arch arteries.

Dermal Lymphatic Capillaries Do Not Obey Murray's Law

Lymphatic vessels serve as a major conduit for the transport of interstitial fluid, immune cells, lipids and drugs. Therefore, increased knowledge about their development and function is relevant to clinical issues ranging from chronic inflammation and edema, to cancer metastasis to targeted drug delivery. Murray's Law is a widely-applied branching rule upheld in diverse circulatory systems including leaf venation, sponge canals, and various human organs for optimal fluid transport. Considering the unique and diverse functions of lymphatic fluid transport, we specifically address the branching of developing lymphatic capillaries, and the flow of lymph through these vessels. Using an empirically-generated dataset from wild type and genetic lymphatic insufficiency mouse models we confirmed that branching blood capillaries consistently follow Murray's Law. However surprisingly, we found that the optimization law for lymphatic vessels follows a different pattern, namely a Murray's Law exponent of ~1.45. In this case, the daughter vessels are smaller relative to the parent than would be predicted by the hypothesized radius-cubed law for impermeable vessels. By implementing a computational fluid dynamics model, we further examined the extent to which the assumptions of Murray's Law were violated. We found that the flow profiles were predominantly parabolic and reasonably followed the assumptions of Murray's Law. These data suggest an alternate hypothesis for optimization of the branching structure of the lymphatic system, which may have bearing on the unique physiological functions of lymphatics compared to the blood vascular system. Thus, it may be the case that the lymphatic branching structure is optimized to enhance lymph mixing, particle exchange, or immune cell transport, which are particularly germane to the use of lymphatics as drug delivery routes.

Refining Our Understanding of the Flow Through Coronary Artery Branches; Revisiting Murray's Law in Human Epicardial Coronary Arteries

Background: Quantification of coronary blood flow is used to evaluate coronary artery disease, but our understanding of flow through branched systems is poor. Murray's law defines coronary morphometric scaling, the relationship between flow (Q) and vessel diameter (D) and is the basis for minimum lumen area targets when intervening on bifurcation lesions. Murray's original law (Q α DP) dictates that the exponent (P) is 3.0, whilst constant blood velocity throughout the system would suggest an exponent of 2.0. In human coronary arteries, the value of Murray's exponent remains unknown. Aim: To establish the exponent in Murray's power law relationship that best reproduces coronary blood flows (Q) and microvascular resistances (Rmicro) in a bifurcating coronary tree. Methods and Results: We screened 48 cases, and were able to evaluate inlet Q and Rmicro in 27 branched coronary arteries, taken from 20 patients, using a novel computational fluid dynamics (CFD) model which reconstructs 3D coronary anatomy from angiography and uses pressure-wire measurements to compute Q and Rmicro distribution in the main- and side-branches. Outputs were validated against invasive measurements using a Rayflow™ catheter. A Murray's power law exponent of 2.15 produced the strongest correlation and closest agreement with inlet Q (zero bias, r = 0.47, p = 0.006) and an exponent of 2.38 produced the strongest correlation and closest agreement with Rmicro (zero bias, r = 0.66, p = 0.0001). Conclusions: The optimal power law exponents for Q and Rmicro were not 3.0, as dictated by Murray's Law, but 2.15 and 2.38 respectively. These data will be useful in assessing patient-specific coronary physiology and tailoring revascularisation decisions.

Lepidoptera demonstrate the relevance of Murray's Law to circulatory systems with tidal flow

BACKGROUND

Murray's Law, which describes the branching architecture of bifurcating tubes, predicts the morphology of vessels in many amniotes and plants. Here, we use insects to explore the universality of Murray's Law and to evaluate its predictive power for the wing venation of Lepidoptera, one of the most diverse insect orders. Lepidoptera are particularly relevant to the universality of Murray's Law because their wing veins have tidal, or oscillatory, flow of air and hemolymph. We examined over one thousand wings representing 667 species of Lepidoptera.

RESULTS

We found that veins with a diameter above approximately 50 microns conform to Murray's Law, with veins below 50 microns in diameter becoming less and less likely to conform to Murray's Law as they narrow. The minute veins that are most likely to deviate from Murray's Law are also the most likely to have atrophied, which prevents efficient fluid transport regardless of branching architecture. However, the veins of many taxa continue to branch distally to the areas where they atrophied, and these too conform to Murray's Law at larger diameters (e.g., Sesiidae).

CONCLUSIONS

This finding suggests that conformity to Murray's Law in larger taxa may reflect requirements for structural support as much as fluid transport, or may indicate that selective pressures for fluid transport are stronger during the pupal stage-during wing development prior to vein atrophy-than the adult stage. Our results increase the taxonomic scope of Murray's Law and provide greater clarity about the relevance of body size.

Transarterial drug delivery for liver cancer: numerical simulations and experimental validation of particle distribution in patient-specific livers

Background: Transarterial therapies are routinely used for the locoregional treatment of unresectable hepatocellular carcinoma (HCC). However, the impact of clinical parameters (i.e. injection location, particle size, particle density etc.) and patient-specific conditions (i.e. hepatic geometry, cancer burden) on the intrahepatic particle distribution (PD) after transarterial injection of embolizing microparticles is still unclear. Computational fluid dynamics (CFD) may help to better understand this impact.Methods: Using CFD, both the blood flow and microparticle mass transport were modeled throughout the 3D-reconstructed arterial vasculature of a patient-specific healthy and cirrhotic liver. An experimental feasibility study was performed to simulate the PD in a 3D-printed phantom of the cirrhotic arterial network.Results: Axial and in-plane injection locations were shown to be effective parameters to steer particles toward tumor tissue in both geometries. Increasing particle size or density made it more difficult for particles to exit the domain. As cancer burden increased, the catheter tip location mattered less. The in vitro study and numerical results confirmed that PD largely mimics flow distribution, but that significant differences are still possible.Conclusions: Our findings highlight that optimal parameter choice can lead to selective targeting of tumor tissue, but that targeting potential highly depends on patient-specific conditions.

Image-based morphometric studies of human coronary artery bifurcations with/without coronary artery disease

It is of great clinical significance to study the relationship between coronary bifurcation's morphometrical feature change and coronary artery disease (CAD) lesion. The purpose of this study is to determine the morphological changes in patients with CAD lesion when compared with non-CAD subjects and to find indicators that may be used for cardiovascular disease diagnosis. Computed tomography angiography images from Southern Chinese populations were used to reconstruct three-dimensional coronary arterial trees. Murray's law was introduced to assess the level of deviation of the realistic vascular networks from their optimal state. The results showed CAD Left had the highest deviation values of ARR (0.2552 ± 0.0071 ) and DERR (0.5059 ± 0.0098 ), while non-CAD Right had the lowest values (ARR : 0.1892 ± 0.0066 and DERR : 0.3733 ± 0.0092 , respectively). Moreover, the slope values of the ratio between D m 3 and D s 3 + D l 3 for non-CAD Left, CAD Left, non-CAD Right, and CAD Right were 0.7428, 0.7004, 0.7628, and 0.7577, respectively. Theoretically, the slope value should equal to 1 when the bifurcation structure is in its optimal state. Therefore, these results indicated that coronary bifurcations with CAD lesion deviated from the optimal structure further than those without CAD lesion and coronary bifurcations in right were closer to the optimal structure than those in left. More importantly, the present study found that DERR and AER depended only on the diseased state, but not age, suggesting that DERR and AER were potentially used as two novel indicators for early CAD diagnosis.

The liver, a functionalized vascular structure

The liver is not only the largest organ in the body but also the one playing one of the most important role in the human metabolism as it is in charge of transforming toxic substances in the body. Understanding the way its blood vasculature works is key. In this work we show that the challenge of predicting the hepatic multi-scale vascular network can be met thanks to the constructal law of design evolution. The work unveils the structure of the liver blood flow architecture as a combination of superimposed tree-shaped networks and porous system. We demonstrate that the dendritic nature of the hepatic artery, portal vein and hepatic vein can be predicted, together with their geometrical features (diameter ratio, duct length ratio) as the entire blood flow architectures follow the principle of equipartition of imperfections. At the smallest scale, the shape of the liver elemental systems-the lobules-is discovered, while their permeability is also predicted. The theory is compared with good agreement to anatomical data from the literature.

A parametric model for studying the aorta hemodynamics by means of the computational fluid dynamics

Perturbed aorta hemodynamics, as for the carotid and the coronary artery, has been identified as potential predicting factor for cardiovascular diseases. In this study, we propose a parametric study based on the computational fluid dynamics with the aim of providing information regarding aortic disease. In particular, the blood flow inside a parametrized aortic arch is computed as a function of morphological changes of baseline aorta geometry. Flow patterns, wall shear stress, time average wall shear stress and oscillatory shear index were calculated during the cardiac cycle. The influence of geometrical changes on the hemodynamics and on these variables was evaluated. The results suggest that the distance between inflow and aortic arch and the angle between aortic arch and descending trunk are the most influencing parameters regarding the WSS-related indices while the effect of the inlet diameter seems limited. In particular, an increase of the aforementioned distance produces a reduction of the spatial distribution of the higher values of the time average wall shear stress and of the oscillatory shear index independently on the other two parameters while an increase of the angle produce an opposite effect. Moreover, as expected, the analysis of the wall shear stress descriptors suggests that the inlet diameter influences only the flow intensity. As conclusion, the proposed parametric study can be used to evaluate the aorta hemodynamics and could be also applied in the future, for analyzing pathological cases and virtual situations, such as pre- and/or post-operative cardiovascular surgical states that present enhanced changes in the aorta morphology yet promoting important variations on the considered indexes.

A microfluidic platform for simultaneous quantification of oxygen-dependent viscosity and shear thinning in sickle cell blood

The pathology of sickle cell disease begins with the polymerization of intracellular hemoglobin under low oxygen tension, which leads to increased blood effective viscosity and vaso-occlusion. However, it has remained unclear how single-cell changes propagate up to the scale of bulk blood effective viscosity. Here, we use a custom microfluidic system to investigate how the increase in the stiffness of individual cells leads to an increase in the shear stress required for the same fluid strain in a suspension of softer cells. We characterize both the shear-rate dependence and the oxygen-tension dependence of the effective viscosity of sickle cell blood, and we assess the effect of the addition of increasing fractions of normal cells whose material properties are independent of oxygen tension, a scenario relevant to the treatment of sickle patients with blood transfusion. For untransfused sickle cell blood, we find an overall increase in effective viscosity at all oxygen tensions and shear rates along with an attenuation in the degree of shear-thinning achieved at the lowest oxygen tensions. We also find that in some cases, even a small fraction of transfused blood cells restores the shape of the shear-thinning relationship, though not the overall baseline effective viscosity. These results suggest that untransfused sickle cell blood will show the most extreme relative rheologic impairment in regions of high shear and that introducing even small fractions of normal blood cells may help retain some shear-thinning capability though without addressing a baseline relative increase in effective viscosity independent of shear.

Combined effect of chronic partial occlusion and orthostatic load on the saphenous vein network: A varicosity model in the rat

Objectives

We tested the combined effects of chronic flow obstacle and gravitation on the saphenous vein network of rats.

Methods

A narrowing clip (500 µm, partial occlusion) was administered on the saphenous vein main branch for 4, 8 and 12 weeks, either separately or in combination with chronic orthostatic load (tilted tube-cages for four weeks). Resulting network changes were studied on plastic casts, by video-microscopy, histochemistry–immunohistochemistry and image analysis.

Results

A rich collateral venous network developed containing newly formed masses of retrograde conducting small veins. Their walls had less dense elastica, less contractile protein, increased cell division activity and macrophage invasion, and were more sensitive to chronic gravitational load.

Conclusions

Hemodynamic disturbance induces remodeling of the saphenous vein network. Walls of veins being in the process of flow-induced morphological remodeling are weak and more sensitive to gravitational load. Reticular vein conglomerates, veins with local dilations, and convoluted courses were observed.

Development of a Customizable Hepatic Arterial Tree and Particle Transport Model for Use in Treatment Planning

Optimal treatment planning for radioembolization of hepatic cancers produces sufficient dose to tumors for control and dose to normal liver parenchyma that is below the threshold for toxicity. The non-uniform distribution of particles in liver microanatomy complicates the planning process as different functional regions receive different doses. Having realistic and patient-specific models of the arterial tree and microsphere trapping would be useful for developing more optimal treatment plans. We propose a macrocell-based growth method to generate models of the hepatic arterial tree from the proper hepatic artery to the terminal arterioles supplying the capillaries in the parenchyma. We show how these trees can be adapted to match patient values of pressure, flow, and vessel diameters while still conforming to laws controlling vessel bifurcation, changes in pressure, and blood flow. We also introduce a method to model particle transport within the tree that accounts for vessel and particle diameter distributions and show the non-uniform microsphere deposition pattern that results. Potential applications include investigating dose heterogeneity and microsphere deposition patterns.

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.

Optimal Branching Structure of Fluidic Networks with Permeable Walls

Biological and engineering studies of Hess-Murray's law are focused on assemblies of tubes with impermeable walls. Blood vessels and airways have permeable walls to allow the exchange of fluid and other dissolved substances with tissues. Should Hess-Murray's law hold for bifurcating systems in which the walls of the vessels are permeable to fluid? This paper investigates the fluid flow in a porous-walled T-shaped assembly of vessels. Fluid flow in this branching flow structure is studied numerically to predict the configuration that provides greater access to the flow. Our findings indicate, among other results, that an asymmetric flow (i.e., breaking the symmetry of the flow distribution) may occur in this symmetrical dichotomous system. To derive expressions for the optimum branching sizes, the hydraulic resistance of the branched system is computed. Here we show the T-shaped assembly of vessels is only conforming to Hess-Murray's law optimum as long as they have impervious walls. Findings also indicate that the optimum relationship between the sizes of parent and daughter tubes depends on the wall permeability of the assembled tubes. Our results agree with analytical results obtained from a variety of sources and provide new insights into the dynamics within the assembly of vessels.

The Humboldt Penguin (Spheniscus humboldti) Rete Tibiotarsale - A supreme biological heat exchanger

Humans are unable to survive low temperature environments without custom designed clothing and support systems. In contrast, certain penguin species inhabit extremely cold climates without losing substantial energy to self-heating (emperor penguins ambient temperature plummets to as low as -45°C). Penguins accomplish this task by relying on distinct anatomical, physiological and behavioral adaptations. One such adaptation is a blood vessel heat exchanger called the 'Rete Tibiotarsale' - an intermingled network of arteries and veins found in penguins' legs. The Rete existence results in blood occupying the foot expressing a lower average temperature and thus the penguin loosing less heat to the ground. This study examines the Rete significance for the species thermal endurance. The penguin anatomy (leg and main blood vessels) is reconstructed using data chiefly based on the Humboldt species. The resulting model is thermally analyzed using finite element (COMSOL) with the species environment used as boundary conditions. A human-like blood vessel configuration, scaled to the penguin's dimensions, is used as a control for the study. Results indicate that the Rete existence facilitates upkeep of 25-65% of the species total metabolic energy production as compared with the human-like configuration; thus making the Rete probably crucial for penguin thermal endurance. Here, we quantitatively link for the first time the function and structure of this remarkable physiological phenotype.

Link between deviations from Murray's Law and occurrence of low wall shear stress regions in the left coronary artery

Murray developed two laws for the geometry of bifurcations in the circulatory system. Based on the principle of energy minimization, Murray found restrictions for the relation between the diameters and also between the angles of the branches. It is known that bifurcations are prone to the development of atherosclerosis, in regions associated to low wall shear stresses (WSS) and high oscillatory shear index (OSI). These indicators (size of low WSS regions, size of high OSI regions and size of high helicity regions) were evaluated in this work. All of them were normalized by the size of the outflow branches. The relation between Murray's laws and the size of low WSS regions was analysed in detail. It was found that the main factor leading to large regions of low WSS is the so called expansion ratio, a relation between the cross section areas of the outflow branches and the cross section area of the main branch. Large regions of low WSS appear for high expansion ratios. Furthermore, the size of low WSS regions is independent of the ratio between the diameters of the outflow branches. Since the expansion ratio in bifurcations following Murray's law is kept in a small range (1 and 1.25), all of them have regions of low WSS with similar size. However, the expansion ratio is not small enough to completely prevent regions with low WSS values and, therefore, Murray's law does not lead to atherosclerosis minimization. A study on the effect of the angulation of the bifurcation suggests that the Murray's law for the angles does not minimize the size of low WSS regions.

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.

Toward an optimal design principle in symmetric and asymmetric tree flow networks

Fluid flow in tree-shaped networks plays an important role in both natural and engineered systems. This paper focuses on laminar flows of Newtonian and non-Newtonian power law fluids in symmetric and asymmetric bifurcating trees. Based on the constructal law, we predict the tree-shaped architecture that provides greater access to the flow subjected to the total network volume constraint. The relationships between the sizes of parent and daughter tubes are presented both for symmetric and asymmetric branching tubes. We also approach the wall-shear stresses and the flow resistance in terms of first tube size, degree of asymmetry between daughter branches, and rheological behavior of the fluid. The influence of tubes obstructing the fluid flow is also accounted for. The predictions obtained by our theory-driven approach find clear support in the findings of previous experimental studies.

Numerical design and optimization of hydraulic resistance and wall shear stress inside pressure-driven microfluidic networks

Microfluidic networks represent the milestone of microfluidic devices. Recent advancements in microfluidic technologies mandate complex designs where both hydraulic resistance and pressure drop across the microfluidic network are minimized, while wall shear stress is precisely mapped throughout the network. In this work, a combination of theoretical and modeling techniques is used to construct a microfluidic network that operates under minimum hydraulic resistance and minimum pressure drop while constraining wall shear stress throughout the network. The results show that in order to minimize the hydraulic resistance and pressure drop throughout the network while maintaining constant wall shear stress throughout the network, geometric and shape conditions related to the compactness and aspect ratio of the parent and daughter branches must be followed. Also, results suggest that while a "local" minimum hydraulic resistance can be achieved for a geometry with an arbitrary aspect ratio, a "global" minimum hydraulic resistance occurs only when the aspect ratio of that geometry is set to unity. Thus, it is concluded that square and equilateral triangular cross-sectional area microfluidic networks have the least resistance compared to all rectangular and isosceles triangular cross-sectional microfluidic networks, respectively. Precise control over wall shear stress through the bifurcations of the microfluidic network is demonstrated in this work. Three multi-generation microfluidic network designs are considered. In these three designs, wall shear stress in the microfluidic network is successfully kept constant, increased in the daughter-branch direction, or decreased in the daughter-branch direction, respectively. For the multi-generation microfluidic network with constant wall shear stress, the design guidelines presented in this work result in identical profiles of wall shear stresses not only within a single generation but also through all the generations of the microfluidic network under investigation. The results obtained in this work are consistent with previously reported data and suitable for a wide range of lab-on-chip applications.

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.

The promise of microfluidic artificial lungs

Microfluidic or microchannel artificial lungs promise to enable a new class of truly portable, therapeutic artificial lungs through feature sizes and blood channel designs that closely mimic those found in their natural counterpart. These new artificial lungs could potentially: 1) have surface areas and priming volumes that are a fraction of current technologies thereby decreasing device size and reducing the foreign body response; 2) contain blood flow networks in which cells and platelets experience pressures, shear stresses, and branching angles that copy those in the human lung thereby improving biocompatibility; 3) operate efficiently with room air, eliminating the need for gas cylinders and complications associated with hyperoxemia; 4) exhibit biomimetic hydraulic resistances, enabling operation with natural pressures and eliminating the need for blood pumps; and, 5) provide increased gas exchange capacity enabling respiratory support for active patients. This manuscript reviews recent research efforts in microfluidic artificial lungs targeted at achieving the advantages above, investigates the ultimate performance and scaling limits of these devices using a proven mathematical model, and discusses the future challenges that must be overcome in order for microfluidic artificial lungs to be applied in the clinic. If all of these promising advantages are realized and the remaining challenges are met, microfluidic artificial lungs could revolutionize the field of pulmonary rehabilitation.

Morphometric, geographic, and territorial characterization of brain arterial trees

Morphometric information of the brain vascularization is valuable for a variety of clinical and scientific applications. In particular, this information is important when creating arterial tree models for imposing boundary conditions in numerical simulations of the brain hemodynamics. The purpose of this work is to provide quantitative descriptions of arterial branches, bifurcation patterns, shape, and geographical distribution of the arborization of the main cerebral arteries as well as estimations of the corresponding vascular territories. For this purpose, subject-specific digital reconstructions of the brain vascular network created from 3T magnetic resonance angiography images of healthy volunteers are used to derive population-averaged morphometric characteristics of the cerebral arterial trees. Copyri

Effect of the degree of LAD stenosis on "competitive flow" and flow field characteristics in LIMA-to-LAD bypass surgery

The long-term patency of the left internal mammary artery (LIMA) in left anterior descending (LAD) coronary stenosis bypass surgery is believed to be related to the degree of competitive flow between the LAD and LIMA. To investigate the effect of the LAD stenosis severity on this phenomenon and on haemodynamics in the LIMA and anastomosis region, a numerical LIMA-LAD model was developed based on 3D geometric (obtained from a cast) and hemodynamic data from an experimental pig study. Proximal LAD pressure was used as upstream boundary condition. The model counted 13 outlets (12 septal arteries and the distal LAD) where flow velocities were imposed in systole, while myocardial conductance was imposed in diastole via an implicit scheme. LAD stenoses of 100 (total occlusion), 90, 75 and 0 % area reduction were constructed. Low degree of LAD stenosis was associated with highly competitive flow and low wall shear stress (WSS) in the LIMA, an unfavourable hemodynamic regime which might contribute to WSS-related remodelling of the LIMA and suboptimal long-term LIMA bypass performance.

Assessment of Energy Requirement for the Retinal Arterial Network in Normal and Hypertensive Subjects

The retinal arterial network structure can be altered by systemic diseases such as hypertension and diabetes. In order to compare the energy requirement for maintaining retinal blood flow and vessel wall metabolism between normal and hypertensive subjects, 3D hypothetical models of a representative retinal arterial bifurcation were constructed based on topological features derived from retinal images. Computational analysis of blood flow was performed, which accounted for the non-Newtonian rheological properties of blood and peripheral vessel resistance. The results suggested that the rate of energy required to maintain the blood flow and wall metabolism is much lower for normal subjects than for hypertensives, with the latter requiring 49.2% more energy for an entire retinal arteriolar tree. Among the several morphological factors, length-to-diameter ratio was found to have the most significant influence on the overall energy requirement.

Clinical benefits of integrating cardiac and vascular models

INTRODUCTION

Recent advances in computational methods and medical imaging techniques have enabled non-invasive exploration of cardiovascular pathologies, from cardiac level to complex arterial networks. The potential of cardiac and vascular modeling in guiding and monitoring therapies could be further extended through the integration of the two systems.

AREAS COVERED

This review includes advances in methods for cardiac electromechanics and vascular flow simulations. The results of a literature search depicting the state of the art in cardiac and vascular modeling are reviewed. The paper goes on to address the benefits and challenges of combined cardiovascular modeling, highlighting the relevance of specific cardiovascular features and implementation. Various alternative approaches and insights on future directions are presented and analyzed with respect to their applicability to clinical practice.

EXPERT OPINION

The article has emerged from the exploration of currently available cardiac and vascular mathematical tools and their corresponding clinical application. The summarized analysis suggests that future efforts should be aimed at developing more accurate and patient-specific mathematical models integrating cardiac and vascular functions to enhance the knowledge of cardiovascular pathologies.

Theoretical models for coronary vascular biomechanics: progress & challenges

A key aim of the cardiac Physiome Project is to develop theoretical models to simulate the functional behaviour of the heart under physiological and pathophysiological conditions. Heart function is critically dependent on the delivery of an adequate blood supply to the myocardium via the coronary vasculature. Key to this critical function of the coronary vasculature is system dynamics that emerge via the interactions of the numerous constituent components at a range of spatial and temporal scales. Here, we focus on several components for which theoretical approaches can be applied, including vascular structure and mechanics, blood flow and mass transport, flow regulation, angiogenesis and vascular remodelling, and vascular cellular mechanics. For each component, we summarise the current state of the art in model development, and discuss areas requiring further research. We highlight the major challenges associated with integrating the component models to develop a computational tool that can ultimately be used to simulate the responses of the coronary vascular system to changing demands and to diseases and therapies.

The effects of hemodynamic force on embryonic development

Blood vessels have long been known to respond to hemodynamic force, and several mechanotransduction pathways have been identified. However, only recently have we begun to understand the effects of hemodynamic force on embryonic development. In this review, we will discuss specific examples illustrating the role of hemodynamic force during the development of the embryo, with particular focus on the development of the vascular system and the morphogenesis of the heart. We will also discuss the important functions served by mechanotransduction and hemodynamic force during placentation, as well as in regulating the maintenance and division of embryonic, hematopoietic, neural, and mesenchymal stem cells. Pathological misregulation of mechanosensitive pathways during pregnancy and embryonic development may contribute to the occurrence of cardiovascular birth defects, as well as to a variety of other diseases, including preeclampsia. Thus, there is a need for future studies focusing on better understanding the physiological effects of hemodynamic force during embryonic development and their role in the pathogenesis of disease.