On the principles of the vascular network branching

We propose an explanation of Murray's law without applying the minimality principles. The model deals with a "delivering" artery system of an organ that is characterized, first, by the space-filling embedding into the organ tissue and, second, by the uniform distribution of the blood pressure drop over it. The latter assumption is justified using the available physiological data and the idea about conditions needed for perfect self-regulation. Based on the two statements we get Murray's law, and so, demonstrate that it can be also regarded as a direct consequence of the organism's capacity for controlling finely the blood flow redistribution over peripheral vascular networks.     

The evaluation of Murray's law in Psilotum nudum (Psilotaceae), an analogue of ancestral vascular plants

Previous work has shown that the xylem of seed plants follows Murray's law when conduits do not provide structural support to the plant. Here, compliance with Murray's law was tested in the stem photosynthesizer Psilotum nudum, a seedless vascular plant. Psilotum nudum was chosen because the central stele does not provide structural support, which means that Murray's law is applicable, and because its simple shoot structure resembles the earliest vascular plants. Murray's law predicts that the sum of the conduit radii cubed (Σr(3)) should decrease in direct proportion with the volume flow rate (Q) to maximize the hydraulic conductance per unit vascular investment. Agreement with Murray's law was assessed by estimating the transpiration rate distal to a cross-section, which should determine Q under steady state conditions, and comparing that with the Σr(3) of that cross-section. As predicted, regressions between the Σr(3) of the cross-section and Q resulted in a linear relationship with a y-intercept that was not different from zero. Two more rigorous statistical tests were also unable to reject Murray's law. Psilotum nudum plants also increased their conductance per investment by having more conduits distally than proximally, which is more efficient hydraulically than equal or declining conduit numbers distally.     

Quantitative assessments of morphological and functional properties of biological trees based on their fractal nature.

The branching systems in our body (vascular and bronchial trees) and those in the environment (plant trees and river systems) are characterized by a fractal nature: the self-similarity in the bifurcation pattern. They increase their branch density toward terminals according to a power function with the exponent called fractal dimension (D). From a stochastic model based-on this feature, we formulated the fractal-based integrals to calculate such morphological parameters as aggregated branch length, surface area, and content volume for any given range of radius (r). It was followed by the derivation of branch number and cross-sectional area, by virtue of the logarithmic sectioning of the r axis and of the branch radius-length relation also given by a power function of r with an exponent (alpha). These derivatives allowed us to quantify various hydrodynamic parameters of vascular and bronchial trees as fluid conduit systems, including the individual branch flow rate, mean flow velocity, wall shear rate and stress, internal pressure, and circumferential tension. The validity of these expressions was verified by comparing the outcomes with actual data measured in vivo in the vascular beds. From additional analyses of the terminal branch number, we found a simple equation relating the exponent (m) of the empirical power law (Murray's so-called cube law) to the other exponents as (m=D+alpha). Finally, allometric studies of mammalian vascular trees revealed uniform and scale-independent distributions of terminal arterioles in organs, which afforded an infarct index, reflecting the severity of tissue damage following arterial infarction.     

Pulsatile blood flow, shear force, energy dissipation and Murray's Law

Background  

Murray's Law states that, when a parent blood vessel branches into daughter vessels, the cube of the radius of the parent vessel is equal to the sum of the cubes of the radii of daughter blood vessels. Murray derived this law by defining a cost function that is the sum of the energy cost of the blood in a vessel and the energy cost of pumping blood through the vessel. The cost is minimized when vessel radii are consistent with Murray's Law. This law has also been derived from the hypothesis that the shear force of moving blood on the inner walls of vessels is constant throughout the vascular system. However, this derivation, like Murray's earlier derivation, is based on the assumption of constant blood flow.     

Engineering of In Vitro 3D Capillary Beds by Self-Directed Angiogenic Sprouting

In recent years, microfluidic systems have been used to study fundamental aspects of angiogenesis through the patterning of single-layered, linear or geometric vascular channels. In vivo, however, capillaries exist in complex, three-dimensional (3D) networks, and angiogenic sprouting occurs with a degree of unpredictability in all x,y,z planes. The ability to generate capillary beds in vitro that can support thick, biological tissues remains a key challenge to the regeneration of vital organs. Here, we report the engineering of 3D capillary beds in an in vitro microfluidic platform that is comprised of a biocompatible collagen I gel supported by a mechanical framework of alginate beads. The engineered vessels have patent lumens, form robust ∼1.5 mm capillary networks across the devices, and support the perfusion of 1 µm fluorescent beads through them. In addition, the alginate beads offer a modular method to encapsulate and co-culture cells that either promote angiogenesis or require perfusion for cell viability in engineered tissue constructs. This laboratory-constructed vascular supply may be clinically significant for the engineering of capillary beds and higher order biological tissues in a scalable and modular manner.     

Helical spring template fabrication of cell‐laden microfluidic hydrogels for tissue engineering

Cell‐laden microfluidic hydrogels find great potential applications in microfluidics, tissue engineering, and drug delivery, due to their ability to control mass transport and cell microenvironment. A variety of methods have been developed to fabricate hydrogels with microfluidic channels, such as molding, bioprinting, and photopatterning. However, the relatively simple structure available and the specific equipment required limit their broad applications in tissue engineering. Here, we developed a simple method to fabricate microfluidic hydrogels with helical microchannels based on a helical spring template. Results from both experimental investigation and numerical modeling revealed a significant enhancement on the perfusion ability and cell viability of helical microfluidic hydrogels compared to those with straight microchannels. The feasibility of such a helical spring template method was also demonstrated for microfluidic hydrogels with complex three‐dimensional channel networks such as branched helical microchannels. The method presented here could potentially facilitate the development of vascular tissue engineering and cell microenvironment engineering. Biotechnol. Bioeng. 2013; 110: 980–989. © 2012 Wiley Periodicals, Inc.     

Microfluidic culture models to study the hydrodynamics of tumor progression and therapeutic response

The integration of tissue engineering strategies with microfluidic technologies has enabled the design of in vitro microfluidic culture models that better adapt to morphological changes in tissue structure and function over time. These biomimetic microfluidic scaffolds accurately mimic native 3D microenvironments, as well as permit precise and simultaneous control of chemical gradients, hydrodynamic stresses, and cellular niches within the system. The recent application of microfluidic in vitro culture models to cancer research offers enormous potential to aid in the development of improved therapeutic strategies by supporting the investigation of tumor angiogenesis and metastasis under physiologically relevant flow conditions. The intrinsic material properties and fluid mechanics of microfluidic culture models enable high‐throughput anti‐cancer drug screening, permit well‐defined and controllable input parameters to monitor tumor cell response to various hydrodynamic conditions or treatment modalities, as well as provide a platform for elucidating fundamental mechanisms of tumor physiology. This review highlights recent developments and future applications of microfluidic culture models to study tumor progression and therapeutic targeting under conditions of hydrodynamic stress relevant to the complex tumor microenvironment. Biotechnol. Biotechnol. Bioeng. 2013; 110: 2063–2072. © 2013 Wiley Periodicals, Inc.     

An optimization principle for vascular radius including the effects of smooth muscle tone.

An optimization principle is proposed for the regulation of vascular morphology. This principle, which extends Murray's law, is based on the hypothesis that blood vessel diameter is controlled by a mechanism that minimizes the total energy required to drive the blood flow, to maintain the blood supply, and to support smooth muscle tone. A theoretical analysis reveals that the proposed principle predicts that the optimum shear stress on the vessel wall due to blood flow increases with blood pressure. This result agrees qualitatively with published findings that the fluid shear stress in veins is significantly smaller than it is in arteries.     

Analysis of intracellular reducing levels in human hepatocytes on three‐dimensional focusing microchip

A novel three‐dimensional hydrodynamic focusing microfluidic device integrated with high‐throughput cell sampling and detection of intracellular contents is presented. It has a pivotal role in maintaining the reducing environment in cells. Intracellular reducing species such as vitamin C and glutathione in normal and tumor cells were labeled by a newly synthesized 2,2,6,6‐tetramethyl‐piperidine‐1‐oxyl‐based fluorescent probe. Hepatocytes are adherent cells, which are prone to attaching to the channel surface. To avoid the attachment of cells on the channel surface, a single channel microchip with three sheath‐flow channels located on both sides of and below the sampling channel was developed. Hydrostatic pressure generated by emptying the sample waste reservoir was used as driving force of fluid on the microchip. Owing to the difference between the liquid levels of the reservoirs, the labeled cells were three‐dimensional hydrodynamically focused and transported from the sample reservoir to the sample waste reservoir. Hydrostatic pressure takes advantage of its ease of generation on a microfluidic chip without any external pressure pump, which drives three sheath‐flow streams to constrain a sample flow stream into a narrow stream to avoid blockage of the sampling channel by adhered cells. The intracellular reducing levels of HepG2 cells and L02 cells were detected by home‐built laser‐induced fluorescence detector. The analysis throughput achieved in this microfluidic system was about 59–68 cells/min. Copyright © 2013 John Wiley & Sons, Ltd.     

A new fundamental bioheat equation for muscle tissue--part II: Temperature of SAV vessels

In this study, a new theoretical framework was developed to investigate temperature variations along countercurrent SAV blood vessels from 300 to 1000 microm diameter in skeletal muscle. Vessels of this size lie outside the range of validity of the Weinbaum-Jiji bioheat equation and, heretofore, have been treated using discrete numerical methods. A new tissue cylinder surrounding these vessel pairs is defined based on vascular anatomy, Murray's law, and the assumption of uniform perfusion. The thermal interaction between the blood vessel pair and surrounding tissue is investigated for two vascular branching patterns, pure branching and pure perfusion. It is shown that temperature variations along these large vessel pairs strongly depend on the branching pattern and the local blood perfusion rate. The arterial supply temperature in different vessel generations was evaluated to estimate the arterial inlet temperature in the modified perfusion source term for the s vessels in Part I of this study. In addition, results from the current research enable one to explore the relative contribution of the SAV vessels and the s vessels to the overall thermal equilibration between blood and tissue.     

Implantable microfluidic device for the formation of three-dimensional vasculature by human endothelial progenitor cells

Vasculogenesis is an important morphogenetic event for vascular tissue engineering and ischemic disease treatment. Stem and progenitor cells can contribute to vasculogenesis via endothelial differentiation and direct participation in blood vessel formation. In this study, we developed an implantable microfluidic device to facilitate formation of three-dimensional (3D) vascular structures by human endothelial progenitor cells (hEPCs). The microfluidic device was made of biodegradable poly(lactic-co-glycolic acid) (PLGA) using a microchannel patterned silicon wafer made by soft lithography. A collagen type I (Col I) hydrogel containing hEPCs filled the microfluidic channels to reconstitute a 3D microenvironment for facilitating vascular structure formation by hEPCs. The device seeded with hEPCs was implanted into the subcutaneous space of athymic mice and retrieved one and four weeks after implantation. Histology and immunohistochemistry revealed that hEPCs formed a 3D capillary network expressing endothelial cell-specific proteins in the channel of the PLGA microfluidic device. This result indicates that a 3D microscale extracellular matrix reconstituted in the microchannel can promote the endothelial differentiation of hEPCs and in turn hEPC-mediated vasculogenesis. The PLGA microfluidic device reported herein may be useful as an implantable tissue-engineering scaffold for vascularized tissue reconstruction and therapeutic angiogenesis.     

Characterization of Leaf-Inspired Microfluidic Chips for Pumpless Fluid Transport

Microfluidic networks are extensively used in miniaturized lab-on-a-chip systems. However, most of the existing micro- channels are simply designed and the corresponding microfluidic systems commonly require external pumps to achieve effec- tive fluid transport. Here we employed microfabrication techniques to replicate naturally-optimized leaf venations into synthetic hydrogels for the fabrication of pumpless microfluidic chips. The unique properties of leaf-inspired microfluidic network in convectively transporting fluid were characterized at different inclination angles. Flow velocity inside these microfluidic net- works was quantitatively measured with Particle Image Velocimetry （PIV）. Mass diffusion from biomimetic microfluidic network to surrounding bulk hydrogels was investigated. The results demonstrate that the leaf-inspired microfluidic network can not only effectively transport fluid without the use of external pumps, but also facilitate rapid mass diffusion within bulk hy- drogel chips. These leaf-inspired microfluidic networks could be potentially used to engineer complex pumpless or- gan-on-a-chip systems.     

Extracellular matrix, mechanotransduction and structural hierarchies in heart tissue engineering

The spatial and temporal scales of cardiac organogenesis and pathogenesis make engineering of artificial heart tissue a daunting challenge. The temporal scales range from nanosecond conformational changes responsible for ion channel opening to fibrillation which occurs over seconds and can lead to death. Spatial scales range from nanometre pore sizes in membrane channels and gap junctions to the metre length scale of the whole cardiovascular system in a living patient. Synchrony over these scales requires a hierarchy of control mechanisms that are governed by a single common principle: integration of structure and function. To ensure that the function of ion channels and contraction of muscle cells lead to changes in heart chamber volume, an elegant choreography of metabolic, electrical and mechanical events are executed by protein networks composed of extracellular matrix, transmembrane integrin receptors and cytoskeleton which are functionally connected across all size scales. These structural control networks are mechanoresponsive, and they process mechanical and chemical signals in a massively parallel fashion, while also serving as a bidirectional circuit for information flow. This review explores how these hierarchical structural networks regulate the form and function of living cells and tissues, as well as how microfabrication techniques can be used to probe this structural control mechanism that maintains metabolic supply, electrical activation and mechanical pumping of heart muscle. Through this process, we delineate various design principles that may be useful for engineering artificial heart tissue in the future.     

A hybrid microfluidic-vacuum device for direct interfacing with conventional cell culture methods

Background

Microfluidics is an enabling technology with a number of advantages over traditional tissue culture methods when precise control of cellular microenvironment is required. However, there are a number of practical and technical limitations that impede wider implementation in routine biomedical research. Specialized equipment and protocols required for fabrication and setting up microfluidic experiments present hurdles for routine use by most biology laboratories.

Results

We have developed and validated a novel microfluidic device that can directly interface with conventional tissue culture methods to generate and maintain controlled soluble environments in a Petri dish. It incorporates separate sets of fluidic channels and vacuum networks on a single device that allows reversible application of microfluidic gradients onto wet cell culture surfaces. Stable, precise concentration gradients of soluble factors were generated using simple microfluidic channels that were attached to a perfusion system. We successfully demonstrated real-time optical live/dead cell imaging of neural stem cells exposed to a hydrogen peroxide gradient and chemotaxis of metastatic breast cancer cells in a growth factor gradient.

Conclusion

This paper describes the design and application of a versatile microfluidic device that can directly interface with conventional cell culture methods. This platform provides a simple yet versatile tool for incorporating the advantages of a microfluidic approach to biological assays without changing established tissue culture protocols.     

A Theoretical Analysis of Water Transport Through Chondrocytes

Because of the avascular nature of adult cartilage, nutrients and waste products are transported to and from the chondrocytes by diffusion and convection through the extracellular matrix. The convective interstitial fluid flow within and around chondrocytes is poorly understood. This theoretical study demonstrates that the incorporation of a semi-permeable membrane when modeling the chondrocyte leads to the following findings: under mechanical loading of an isolated chondrocyte the intracellular fluid pressure is on the order of tens of Pascals and the transmembrane fluid outflow, on the order of picometers per second, takes several days to subside; consequently, the chondrocyte behaves practically as an incompressible solid whenever the loading duration is on the order of minutes or hours. When embedded in its extracellular matrix (ECM), the chondrocyte response is substantially different. Mechanical loading of the tissue leads to a fluid pressure difference between intracellular and extracellular compartments on the order of tens of kilopascals and the transmembrane outflow, on the order of a nanometer per second, subsides in about 1 h. The volume of the chondrocyte decreases concomitantly with that of the ECM. The interstitial fluid flow in the extracellular matrix is directed around the cell, with peak values on the order of tens of nanometers per second. The viscous fluid shear stress acting on the cell surface is several orders of magnitude smaller than the solid matrix shear stresses resulting from the ECM deformation. These results provide new insight toward our understanding of water transport in chondrocytes.     

The impact of simplified boundary conditions and aortic arch inclusion on CFD simulations in the mouse aorta: a comparison with mouse-specific reference data

Computational fluid dynamics (CFD) simulations allow for calculation of a detailed flow field in the mouse aorta and can thus be used to investigate a potential link between local hemodynamics and disease development. To perform these simulations in a murine setting, one often needs to make assumptions (e.g. when mouse-specific boundary conditions are not available), but many of these assumptions have not been validated due to a lack of reference data. In this study, we present such a reference data set by combining high-frequency ultrasound and contrast-enhanced micro-CT to measure (in vivo) the time-dependent volumetric flow waveforms in the complete aorta (including seven major side branches) of 10 male ApoE -/- deficient mice on a C57Bl/6 background. In order to assess the influence of some assumptions that are commonly applied in literature, four different CFD simulations were set up for each animal: (i) imposing the measured volumetric flow waveforms, (ii) imposing the average flow fractions over all 10 animals, presented as a reference data set, (iii) imposing flow fractions calculated by Murray's law, and (iv) restricting the geometrical model to the abdominal aorta (imposing measured flows). We found that - even if there is sometimes significant variation in the flow fractions going to a particular branch - the influence of using average flow fractions on the CFD simulations is limited and often restricted to the side branches. On the other hand, Murray's law underestimates the fraction going to the brachiocephalic trunk and strongly overestimates the fraction going to the distal aorta, influencing the outcome of the CFD results significantly. Changing the exponential factor in Murray's law equation from 3 to 2 (as suggested by several authors in literature) yields results that correspond much better to those obtained imposing the average flow fractions. Restricting the geometrical model to the abdominal aorta did not influence the outcome of the CFD simulations. In conclusion, the presented reference dataset can be used to impose boundary conditions in the mouse aorta in future studies, keeping in mind that they represent a subsample of the total population, i.e., relatively old, non-diseased, male C57Bl/6 ApoE -/- mice.     

Fabrication of disposable protein chip for simultaneous sample detection

In this study, we have described a method for the fabrication of a protein chip on silicon substrate using hydrophobic thin film and microfluidic channels, for the simultaneous detection of multiple targets in samples. The use of hydrophobic thin film provides for a physical, chemical, and biological barrier for protein patterning. The microfluidic channels create four protein patterned strips on the silicon surfaces with a high signal-to-noise ratio. The feasibility of the protein chips was determined in order to discriminate between each protein interaction in a mixture sample that included biotin, ovalbumin, hepatitis B antigen. In the fabrication of the multiplexed assay system, the utilization of the hydrophobic thin film and the microfluidic networks constitutes a more convenient method for the development of biosensors or biochips. This technique may be applicable to the simultaneous evaluation of multiple protein-protein interactions.     

Successful development of small diameter tissue-engineering vascular vessels by our novel integrally designed pulsatile perfusion-based bioreactor

Small-diameter (<4 mm) vascular constructs are urgently needed for patients requiring replacement of their peripheral vessels. However, successful development of constructs remains a significant challenge. In this study, we successfully developed small-diameter vascular constructs with high patency using our integrally designed computer-controlled bioreactor system. This computer-controlled bioreactor system can confer physiological mechanical stimuli and fluid flow similar to physiological stimuli to the cultured grafts. The medium circulating system optimizes the culture conditions by maintaining fixed concentration of O(2) and CO(2) in the medium flow and constant delivery of nutrients and waste metabolites, as well as eliminates the complicated replacement of culture medium in traditional vascular tissue engineering. Biochemical and mechanical assay of newly developed grafts confirm the feasibility of the bioreactor system for small-diameter vascular engineering. Furthermore, the computer-controlled bioreactor is superior for cultured cell proliferation compared with the traditional non-computer-controlled bioreactor. Specifically, our novel bioreactor system may be a potential alternative for tissue engineering of large-scale small-diameter vascular vessels for clinical use.     

Microfluidic cell culture models for tissue engineering

Microfluidic systems have emerged as revolutionary new platform technologies for a range of applications, from consumer products such as inkjet printer cartridges to lab-on-a-chip diagnostic systems. Recent developments have opened the door to a new set of opportunities for microfluidic systems, in the field of tissue and organ engineering. Advances in the design of physiologically relevant structures and networks, fabrication processes for biomaterials suitable for in vivo use, and techniques for scaling towards large, three-dimensional constructs, are converging towards therapeutic applications of microfluidic technologies in engineering complex tissues and organs. These advances herald a new generation of microfluidics-based approaches designed for specific tissue and organ applications, incorporating microvascular networks, structures for transport and filtration, and a three-dimensional microenvironment suitable for supporting phenotypic cell behavior, tissue function, and implantation and host integration.     

The influence of boundary conditions on wall shear stress distribution in patients specific coronary trees

Patient specific geometrical data on human coronary arteries can be reliably obtained multislice computer tomography (MSCT) imaging. MSCT cannot provide hemodynamic variables, and the outflow through the side branches must be estimated. The impact of two different models to determine flow through the side branches on the wall shear stress (WSS) distribution in patient specific geometries is evaluated. Murray's law predicts that the flow ratio through the side branches scales with the ratio of the diameter of the side branches to the third power. The empirical model is based on flow measurements performed by Doriot et al. (2000) in angiographically normal coronary arteries. The fit based on these measurements showed that the flow ratio through the side branches can best be described with a power of 2.27. The experimental data imply that Murray's law underestimates the flow through the side branches. We applied the two models to study the WSS distribution in 6 coronary artery trees. Under steady flow conditions, the average WSS between the side branches differed significantly for the two models: the average WSS was 8% higher for Murray's law and the relative difference ranged from -5% to +27%. These differences scale with the difference in flow rate. Near the bifurcations, the differences in WSS were more pronounced: the size of the low WSS regions was significantly larger when applying the empirical model (13%), ranging from -12% to +68%. Predicting outflow based on Murray's law underestimates the flow through the side branches. Especially near side branches, the regions where atherosclerotic plaques preferentially develop, the differences are significant and application of Murray's law underestimates the size of the low WSS region.