Minimum mass vascular networks in multifunctional materials

Using DFT on ultrasound measurements to determine patient-specific blood flow boundary conditions for computational hemodynamics of intracranial aneurysms

Boundary conditions (BCs) is one pivotal factor influencing the accuracy of hemodynamic predictions on intracranial aneurysms (IAs) using computational fluid dynamics (CFD) modeling. Unfortunately, a standard procedure to secure accurate BCs for hemodynamic modeling does not exist. To bridge such a knowledge gap, two representative patient-specific IA models (Case-I and Case-II) were reconstructed and their blood flow velocity waveforms in the internal carotid artery (ICA) were measured by ultrasonic techniques and modeled by discrete Fourier transform (DFT). Then, numerical investigations were conducted to explore the appropriate number of samples (N) for DFT modeling to secure the accurate BC by comparing a series of hemodynamic parameters using in-vitro validated CFD modeling. Subsequently, a comprehensive comparison in hemodynamic characteristics under patient-specific BCs and a generalized BC based on a one-dimensional (1D) model was conducted to reinforce the understanding that a patient-specific BC is pivotal for accurate hemodynamic risk evaluations on IA pathophysiology. In addition, the influence of the variance of heart rate/cardiac pulsatile period on hemodynamic characteristics in IA models was studied preliminarily. The results showed that N ≥ 16 for DFT model is a decent choice to secure the proper BC profile to calculate time-averaged hemodynamic parameters, while more data points such as N ≥ 36 can ensure the accuracy of instantaneous hemodynamic predictions. In addition, results revealed the generalized BC could overestimate or underestimate the hemodynamic risks on IAs significantly; thus, patient-specific BCs are highly recommended for hemodynamic modeling for IA risk evaluation. Furthermore, this study discovered the variance of heart rate has rare influences on hemodynamic characteristics in both instantaneous and time-averaged parameters under the assumption of an identical blood flow rate.

Tuning the Pennes Perfusion Rate to Model Large Vessel Cooling Effects in Hepatic Radiofrequency Ablation

Radio-frequency ablation (RFA) has become a popular method for the minimally invasive treatment of liver cancer. However, the success rate of these treatments depends heavily on the amount of experience the clinician possesses. Mathematical modelling can help mitigate this problem by providing an indication of the treatment outcome. Thermal lesions in RFA are affected by the cooling effect of both fine-scale and large-scale blood vessels. The exact model for large-scale blood vessels is advection-diffusion, i.e. a model capable of producing directional effects, which are known to occur in certain cases. In previous research, in situations where directional effects do not occur, the advection term in the blood vessel model has been typically replaced with the Pennes perfusion term, albeit with a higher-than usual perfusion rate. Whether these values of the perfusion rate appearing in literature are optimal for the particular vessel radii in question, has not been investigated so far. The present work aims to address this issue. An attempt has been made to determine, for values of vessel radius between 0.55 mm and 5 mm, best estimates for the perfusion rate which minimize the error in thermal lesion volumes between the perfusion-based model and the advection-based model. The results for the best estimate of the perfusion rate presented may be used in existing methods for fast estimation of RFA outcomes. Furthermore, the possible improvements to the presented methodology have been highlighted.

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.

Study of flow effects on temperature-controlled radiofrequency ablation using phantom experiments and forward simulations

PURPOSE

Blood flow is known to add variability to hepatic radiofrequency ablation (RFA) treatment outcomes. However, few studies exist on its impact on temperature-controlled RFA. Hence, we investigate large-scale blood flow effects on temperature-controlled RFA in flow channel experiments and numerical simulations.

METHODS

Ablation zones were induced in tissue-mimicking, thermochromic phantoms with a single flow channel, using an RF generator with temperature-controlled power delivery and a monopolar needle electrode. Channels were generated by molding the phantom around a removable rod. Channel radius and saline flow rate were varied to study the impact of flow on (i) the ablated cross-sectional area, (ii) the delivered generator power, and (iii) the occurrence of directional effects on the thermal lesion. Finite volume simulations reproducing the experimental geometry, flow conditions, and generator power input were conducted and compared to the experimental ablation outcomes.

RESULTS

Vessels of different channel radii r affected the ablation outcome in different ways. For r = 0.275  mm, the ablated area decreased with increasing flow rate while the energy input was hardly affected. For r = 0.9  mm and r = 2.3  mm, the energy input increased toward larger flow rates; for these radii, the ablated area decreased and increased toward larger flow rates, respectively, while still being reduced overall as compared to the reference experiment without flow. Directional effects, that is, local shrinking of the lesion upstream of the needle and an extension thereof downstream, were observed only for the smallest radius. The simulations qualitatively confirmed these observations. As compared to performing the simulations without flow, including flow effects in the simulations reduced the mean absolute error between experimental and simulated ablated areas from 0.23 to 0.12.

CONCLUSION

While the temperature control mechanism did not detect the heat sink effect in the case of the smallest channel radius, it counteracted the heat sink effect in the case of the larger channel radii with an increased energy input; this explains the increase in ablated area toward high flow rates (for r = 2.3  mm). The experiments in a simple phantom setup, thus, contribute to a good understanding of the phenomenon and are suitable for model validation.

Simulation study of the cooling effect of blood vessels and blood coagulation in hepatic radio-frequency ablation

PURPOSE

Computer simulations of hepatic radio-frequency ablation (RFA) were performed to: (i) determine the dependence of the vessel wall heat transfer coefficient on geometrical parameters; (ii) study the conditions required for the occurrence of the directional effect of blood; and (iii) classify blood vessels according to their effect on the thermal lesion while considering blood coagulation. The information thus obtained supports the development of a multi-scale bio-heat model tailored for more accurate prediction of hepatic RFA outcomes in the vicinity of blood vessels.

MATERIALS AND METHODS

The simulation geometry consisted of healthy tissue, tumor tissue, a mono-polar RF-needle, and a single cylindrical blood vessel. The geometrical parameters of interest were the RF-needle active length and those describing blood vessel configuration. A simple, novel method to incorporate the effects of blood coagulation into the simulation was developed and tested.

RESULTS

A closed form expression giving the dependence of the vessel wall heat transfer coefficient on geometrical parameters was obtained. Directional effects on the thermal lesion were found to occur for blood vessel radii between 0.4 mm and 0.5 mm. Below 0.4 mm blood coagulation blocked the flow.

CONCLUSIONS

The closed form expression for the heat transfer coefficient can be used in models of RFA to speed up computation. The conditions on vessel radii required for the occurrence of directional effects on the thermal lesion were determined. These conditions allow the classification of blood vessels. Different approximations to the thermal equation can thus be used for these vessel classes.

Blood flow rate estimation in optic disc capillaries and vessels using Doppler optical coherence tomography with 3D fast phase unwrapping

The retinal volumetric flow rate contains useful information not only for ophthalmology but also for the diagnosis of common civilization diseases such as diabetes, Alzheimer's disease, or cerebrovascular diseases. Non-invasive optical methods for quantitative flow assessment, such as Doppler optical coherence tomography (OCT), have certain limitations. One is the phase wrapping that makes simultaneous calculations of the flow in all human retinal vessels impossible due to a very large span of flow velocities. We demonstrate that three-dimensional Doppler OCT combined with three-dimensional four Fourier transform fast phase unwrapping (3D 4FT FPU) allows for the calculation of the volumetric blood flow rate in real-time by the implementation of the algorithms in a graphics processing unit (GPU). The additive character of the flow at the furcations is proven using a microfluidic device with controlled flow rates as well as in the retinal veins bifurcations imaged in the optic disc area of five healthy volunteers. We show values of blood flow rates calculated for retinal capillaries and vessels with diameters in the range of 12-150 µm. The potential of quantitative measurement of retinal blood flow volume includes noninvasive detection of carotid artery stenosis or occlusion, measuring vascular reactivity and evaluation of vessel wall stiffness.

Understanding nanoparticle flow with a new in vitro experimental and computational approach using hydrogel channels

Nanoparticles (NPs) are considered as one of the most promising drug delivery vehicles and a next-generation solution for current medical challenges. In this context, variables related to flow of NPs such as the quantity of NPs lost during transport and flow trajectory greatly affect the clinical efficiency of NP drug delivery systems. Currently, there is little knowledge of the physical mechanisms dominating NP flow inside the human body due to the limitations of available experimental tools for mimicking complex physiological environments at the preclinical stage. Here, we report a coupled experimental and computational fluid dynamics (CFD)-based novel in vitro approach to predict the flow velocity and binding of NP drug delivery systems during transport through vasculature. Poly(hydroxyethyl)methacrylate hydrogels were used to form soft cylindrical constructs mimicking vascular sections as flow channels for synthesized iron oxide NPs in these first-of-its-kind transport experiments. Brownian dynamics and material of the flow channels played key roles in NP flow, based on the measurements of NP flow velocity over seven different mass concentrations. A fully developed laminar flow of the NPs under these conditions was simultaneously predicted using CFD. Results from the mass loss of NPs during flow indicated a diffusion-dominated flow at higher particle concentrations but a flow controlled by the surrounding fluid and Brownian dynamics at the lowest NP concentrations. The CFD model predicted a mass loss of 1.341% and 6.253% for the 4.12 g·mL^-1 and 2.008 g·mL^-1 inlet mass concentrations of the NPs, in close confirmation with the experimental results. This further highlights the reliability of our new in vitro technique in providing mechanistic insights of NP flow for potential preclinical stage applications.

Multiparametric topological analysis of reconstructed rat brain arterial system

All metabolic processes in living tissues are provided by the vascular system, whose functionality largely depends on its structure and topology. The paper proposes an algorithm for constructing the circulatory system based on a combination of stochastic and deterministic approaches. Analyses of the topological characteristics of arterial tree models with different values of the bifurcation exponent ([Formula: see text]) and length coefficient ([Formula: see text]) show that the maximum agreement with experimental data can be achieved only with the optimal values of both parameters (3.0 and 0.90, respectively). Application of the multiparametric optimization in conjunction with topological analysis makes it possible to quantify the biological division of the distributing and delivering vessels of a tree with a high degree of branching. The proposed approach allows both the correct spatial localization of the main arteries and the complex topology of the complete arterial system down to the capillaries to be reproduced.

Pruning to Increase Taylor Dispersion in Physarum polycephalum Networks

How do the topology and geometry of a tubular network affect the spread of particles within fluid flows? We investigate patterns of effective dispersion in the hierarchical, biological transport network formed by Physarum polycephalum. We demonstrate that a change in topology-pruning in the foraging state-causes a large increase in effective dispersion throughout the network. By comparison, changes in the hierarchy of tube radii result in smaller and more localized differences. Pruned networks capitalize on Taylor dispersion to increase the dispersion capability.

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.

Cerebral oxygenation and optimal vascular brain organization

The cerebral vascular network has evolved in such a way so as to minimize transport time and energy expenditure. This is accomplished by a subtle combination of the optimal arrangement of arteries, arterioles and capillaries and the transport mechanisms of convection and diffusion. Elucidating the interaction between cerebral vascular architectonics and the latter physical mechanisms can catalyse progress in treating cerebral pathologies such as stroke, brain tumours, dementia and targeted drug delivery. Here, we show that brain microvascular organization is predicated on commensurate intracapillary oxygen convection and parenchymal diffusion times. Cross-species grey matter results for the rat, cat, rabbit and human reveal very good correlation between the cerebral capillary and tissue mean axial oxygen convective and diffusion time intervals. These findings agree with the constructal principle.

Obstructions in Vascular Networks: Relation Between Network Morphology and Blood Supply

We relate vascular network structure to hemodynamics after vessel obstructions. We consider tree-like networks with a viscoelastic fluid with the rheological characteristics of blood. We analyze the network hemodynamic response, which is a function of the frequencies involved in the driving, and a measurement of the resistance to flow. This response function allows the study of the hemodynamics of the system, without the knowledge of a particular pressure gradient. We find analytical expressions for the network response, which explicitly show the roles played by the network structure, the degree of obstruction, and the geometrical place in which obstructions occur. Notably, we find that the sequence of resistances of the network without occlusions strongly determines the tendencies that the response function has with the anatomical place where obstructions are located. We identify anatomical sites in a network that are critical for its overall capacity to supply blood to a tissue after obstructions. We demonstrate that relatively small obstructions in such critical sites are able to cause a much larger decrease on flow than larger obstructions placed in non-critical sites. Our results indicate that, to a large extent, the response of the network is determined locally. That is, it depends on the structure that the vasculature has around the place where occlusions are found. This result is manifest in a network that follows Murray’s law, which is in reasonable agreement with several mammalian vasculatures. For this one, occlusions in early generation vessels have a radically different effect than occlusions in late generation vessels occluding the same percentage of area available to flow. This locality implies that whenever there is a tissue irrigated by a tree-like in vivo vasculature, our model is able to interpret how important obstructions are for the irrigation of such tissue.

Constructal law of vascular trees for facilitation of flow

Diverse tree structures such as blood vessels, branches of a tree and river basins exist in nature. The constructal law states that the evolution of flow structures in nature has a tendency to facilitate flow. This study suggests a theoretical basis for evaluation of flow facilitation within vascular structure from the perspective of evolution. A novel evolution parameter (Ev) is proposed to quantify the flow capacity of vascular structures. Ev is defined as the ratio of the flow conductance of an evolving structure (configuration with imperfection) to the flow conductance of structure with least imperfection. Attaining higher Ev enables the structure to expedite flow circulation with less energy dissipation. For both Newtonian and non-Newtonian fluids, the evolution parameter was developed as a function of geometrical shape factors in laminar and turbulent fully developed flows. It was found that the non-Newtonian or Newtonian behavior of fluid as well as flow behavior such as laminar or turbulent behavior affects the evolution parameter. Using measured vascular morphometric data of various organs and species, the evolution parameter was calculated. The evolution parameter of the tree structures in biological systems was found to be in the range of 0.95 to 1. The conclusion is that various organs in various species have high capacity to facilitate flow within their respective vascular structures.

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.

Nanocarrier Hydrodynamics and Binding in Targeted Drug Delivery: Challenges in Numerical Modeling and Experimental Validation

This review discusses current progress and future challenges in the numerical modeling of targeted drug delivery using functionalized nanocarriers (NC). Antibody coated nanocarriers of various size and shapes, also called functionalized nanocarriers, are designed to be injected in the vasculature, whereby they undergo translational and rotational motion governed by hydrodynamic interaction with blood particulates as well as adhesive interactions mediated by the surface antibody binding to target antigens/receptors on cell surfaces. We review current multiscale modeling approaches rooted in computational fluid dynamics and nonequilibrium statistical mechanics to accurately resolve fluid, thermal, as well as adhesive interactions governing nanocarrier motion and their binding to endothelial cells lining the vasculature. We also outline current challenges and unresolved issues surrounding the modeling methods. Experimental approaches in pharmacology and bioengineering are discussed briefly from the perspective of model validation.

Biomimetic reliability strategies for self-healing vascular networks in engineering materials

Self-healing via a vascular network is an active research topic, with several recent publications reporting the application and optimization of these systems. This work represents the first consideration of the probable failure modes of a self-healing system as a driver for network design. The critical failure modes of a proposed self-healing system based on a vascular network were identified via a failure modes, effects and criticality analysis and compared to those of the human circulatory system. A range of engineering and biomimetic design concepts to address these critical failure modes is suggested with minimum system mass the overall design driver for high-performance systems. Plant vasculature has been mimicked to propose a segregated network to address the risk of fluid leakage. This approach could allow a network to be segregated into six separate paths with a system mass penalty of only approximately 25%. Fluid flow interconnections that mimic the anastomoses of animal vasculatures can be used within a segregated network to balance the risk of failure by leakage and blockage. These biomimetic approaches define a design space that considers the existing published literature in the context of system reliability.