Red blood cells and other non-spherical capsules in shear flow: oscillatory dynamics and the tank-treading-to-tumbling transition

We consider the motion of red blood cells and other non-spherical microcapsules dilutely suspended in a simple shear flow. Our analysis indicates that depending on the viscosity, membrane elasticity, geometry and shear rate, the particle exhibits either tumbling, tank-treading of the membrane about the viscous interior with periodic oscillations of the orientation angle, or intermittent behavior in which the two modes occur alternately. For red blood cells, we compute the complete phase diagram and identify a novel tank-treading-to-tumbling transition at low shear rates. Observations of such motions coupled with our theoretical framework may provide a sensitive means of assessing capsule properties.Comment: 11 pages, 4 figure

A simplified particulate model for coarse-grained hemodynamics simulations

Human blood flow is a multi-scale problem: in first approximation, blood is a dense suspension of plasma and deformable red cells. Physiological vessel diameters range from about one to thousands of cell radii. Current computational models either involve a homogeneous fluid and cannot track particulate effects or describe a relatively small number of cells with high resolution, but are incapable to reach relevant time and length scales. Our approach is to simplify much further than existing particulate models. We combine well established methods from other areas of physics in order to find the essential ingredients for a minimalist description that still recovers hemorheology. These ingredients are a lattice Boltzmann method describing rigid particle suspensions to account for hydrodynamic long range interactions and---in order to describe the more complex short-range behavior of cells---anisotropic model potentials known from molecular dynamics simulations. Paying detailedness, we achieve an efficient and scalable implementation which is crucial for our ultimate goal: establishing a link between the collective behavior of millions of cells and the macroscopic properties of blood in realistic flow situations. In this paper we present our model and demonstrate its applicability to conditions typical for the microvasculature.Comment: 12 pages, 11 figure

Soft lubrication: the elastohydrodynamics of non-conforming and conforming contacts

We study the lubrication of fluid-immersed soft interfaces and show that elastic deformation couples tangential and normal forces and thus generates lift. We consider materials that deform easily, due to either geometry (e.g. a shell) or constitutive properties (e.g. a gel or a rubber), so that the effects of pressure and temperature on the fluid properties may be neglected. Four different system geometries are considered: a rigid cylinder moving parallel to a soft layer coating a rigid substrate; a soft cylinder moving parallel to a rigid substrate; a cylindrical shell moving parallel to a rigid substrate; and finally a cylindrical conforming journal bearing coated with a thin soft layer. In addition, for the particular case of a soft layer coating a rigid substrate we consider both elastic and poroelastic material responses. For all these cases we find the same generic behavior: there is an optimal combination of geometric and material parameters that maximizes the dimensionless normal force as a function of the softness parameter = hydrodynamic pressure/elastic stiffness = surface deflection/gap thickness which characterizes the fluid-induced deformation of the interface. The corresponding cases for a spherical slider are treated using scaling concepts.Comment: 61 pages, 20 figures, 2 tables, submitted to Physics of Fluid

Multiscale modelling of vascular tumour growth in 3D: the roles of domain size & boundary condition

We investigate a three-dimensional multiscale model of vascular tumour growth, which couples blood flow, angiogenesis, vascular remodelling, nutrient/growth factor transport, movement of, and interactions between, normal and tumour cells, and nutrient-dependent cell cycle dynamics within each cell. In particular, we determine how the domain size, aspect ratio and initial vascular network influence the tumour's growth dynamics and its long-time composition. We establish whether it is possible to extrapolate simulation results obtained for small domains to larger ones, by constructing a large simulation domain from a number of identical subdomains, each subsystem initially comprising two parallel parent vessels, with associated cells and diffusible substances. We find that the subsystem is not representative of the full domain and conclude that, for this initial vessel geometry, interactions between adjacent subsystems contribute to the overall growth dynamics. We then show that extrapolation of results from a small subdomain to a larger domain can only be made if the subdomain is sufficiently large and is initialised with a sufficiently complex vascular network. Motivated by these results, we perform simulations to investigate the tumour's response to therapy and show that the probability of tumour elimination in a larger domain can be extrapolated from simulation results on a smaller domain. Finally, we demonstrate how our model may be combined with experimental data, to predict the spatio-temporal evolution of a vascular tumour

Toward an Ising Model of Cancer and Beyond

Theoretical and computational tools that can be used in the clinic to predict neoplastic progression and propose individualized optimal treatment strategies to control cancer growth is desired. To develop such a predictive model, one must account for the complex mechanisms involved in tumor growth. Here we review resarch work that we have done toward the development of an "Ising model" of cancer. The review begins with a description of a minimalist four-dimensional (three in space and one in time) cellular automaton (CA) model of cancer in which healthy cells transition between states (proliferative, hypoxic, and necrotic) according to simple local rules and their present states, which can viewed as a stripped-down Ising model of cancer. This model is applied to model the growth of glioblastoma multiforme, the most malignant of brain cancers. This is followed by a discussion of the extension of the model to study the effect on the tumor dynamics and geometry of a mutated subpopulation. A discussion of how tumor growth is affected by chemotherapeutic treatment is then described. How angiogenesis as well as the heterogeneous and confined environment in which a tumor grows is incorporated in the CA model is discussed. The characterization of the level of organization of the invasive network around a solid tumor using spanning trees is subsequently described. Then, we describe open problems and future promising avenues for future research, including the need to develop better molecular-based models that incorporate the true heterogeneous environment over wide range of length and time scales (via imaging data), cell motility, oncogenes, tumor suppressor genes and cell-cell communication. The need to bring to bear the powerful machinery of the theory of heterogeneous media to better understand the behavior of cancer in its microenvironment is presented.Comment: 55 pages, 21 figures and 3 tables. To appear in Physical Biology. Added reference

Multi-Particle Collision Dynamics -- a Particle-Based Mesoscale Simulation Approach to the Hydrodynamics of Complex Fluids

In this review, we describe and analyze a mesoscale simulation method for fluid flow, which was introduced by Malevanets and Kapral in 1999, and is now called multi-particle collision dynamics (MPC) or stochastic rotation dynamics (SRD). The method consists of alternating streaming and collision steps in an ensemble of point particles. The multi-particle collisions are performed by grouping particles in collision cells, and mass, momentum, and energy are locally conserved. This simulation technique captures both full hydrodynamic interactions and thermal fluctuations. The first part of the review begins with a description of several widely used MPC algorithms and then discusses important features of the original SRD algorithm and frequently used variations. Two complementary approaches for deriving the hydrodynamic equations and evaluating the transport coefficients are reviewed. It is then shown how MPC algorithms can be generalized to model non-ideal fluids, and binary mixtures with a consolute point. The importance of angular-momentum conservation for systems like phase-separated liquids with different viscosities is discussed. The second part of the review describes a number of recent applications of MPC algorithms to study colloid and polymer dynamics, the behavior of vesicles and cells in hydrodynamic flows, and the dynamics of viscoelastic fluids

Quantification of longitudinal tissue pO2 gradients in window chamber tumours: impact on tumour hypoxia

We previously reported that the arteriolar input in window chamber tumours is limited in number and is constrained to enter the tumour from one surface, and that the pO2 of tumour arterioles is lower than in comparable arterioles of normal tissues. On average, the vascular pO2 in vessels of the upper surface of these tumours is lower than the pO2 of vessels on the fascial side, suggesting that there may be steep vascular longitudinal gradients (defined as the decline in vascular pO2 along the afferent path of blood flow) that contribute to vascular hypoxia on the upper surface of the tumours. However, we have not previously measured tissue pO2 on both surfaces of these chambers in the same tumour. In this report, we investigated the hypothesis that the anatomical constraint of arteriolar supply from one side of the tumour results in longitudinal gradients in pO2 sufficient in magnitude to create vascular hypoxia in tumours grown in dorsal flap window chambers. Fischer-344 rats had dorsal flap window chambers implanted in the skin fold with simultaneous transplantation of the R3230AC tumour. Tumours were studied at 9–11 days after transplantation, at a diameter of 3–4 mm; the tissue thickness was 200 μm. For magnetic resonance microscopic imaging, gadolinium DTPA bovine serum albumin (BSA-DTPA-Gd) complex was injected i.v., followed by fixation in 10% formalin and removal from the animal. The sample was imaged at 9.4 T, yielding voxel sizes of 40 μm. Intravital microscopy was used to visualize the position and number of arterioles entering window chamber tumour preparations. Phosphorescence life time imaging (PLI) was used to measure vascular pO2. Blue and green light excitations of the upper and lower surfaces of window chambers were made (penetration depth of light ~50 vs >200 μm respectively). Arteriolar input into window chamber tumours was limited to 1 or 2 vessels, and appeared to be constrained to the fascial surface upon which the tumour grows. PLI of the tumour surface indicated greater hypoxia with blue compared with green light excitation (P < 0.03 for 10th and 25th percentiles and for per cent pixels < 10 mmHg). In contrast, illumination of the fascial surface with blue light indicated less hypoxia compared with illumination of the tumour surface (P < 0.05 for 10th and 25th percentiles and for per cent pixels < 10 mmHg). There was no significant difference in pO2 distributions for blue and green light excitation from the fascial surface nor for green light excitation when viewed from either surface. The PLI data demonstrates that the upper surface of the tumour is more hypoxic because blue light excitation yields lower pO2 values than green light excitation. This is further verified in the subset of chambers in which blue light excitation of the fascial surface showed higher pO2 distributions compared with the tumour surface. These results suggest that there are steep longitudinal gradients in vascular pO2 in this tumour model that are created by the limited number and orientation of the arterioles. This contributes to tumour hypoxia. Arteriolar supply is often limited in other tumours as well, suggesting that this may represent another cause for tumour hypoxia. This report is the first direct demonstration that longitudinal oxygen gradients actually lead to hypoxia in tumours. © 1999 Cancer Research Campaig

Vesicle-Like Biomechanics Governs Important Aspects of Nuclear Geometry in Fission Yeast

It has long been known that during the closed mitosis of many unicellular eukaryotes, including the fission yeast (Schizosaccharomyces pombe), the nuclear envelope remains intact while the nucleus undergoes a remarkable sequence of shape transformations driven by elongation of an intranuclear mitotic spindle whose ends are capped by spindle pole bodies embedded in the nuclear envelope. However, the mechanical basis of these normal cell cycle transformations, and abnormal nuclear shapes caused by intranuclear elongation of microtubules lacking spindle pole bodies, remain unknown. Although there are models describing the shapes of lipid vesicles deformed by elongation of microtubule bundles, there are no models describing normal or abnormal shape changes in the nucleus. We describe here a novel biophysical model of interphase nuclear geometry in fission yeast that accounts for critical aspects of the mechanics of the fission yeast nucleus, including the biophysical properties of lipid bilayers, forces exerted on the nuclear envelope by elongating microtubules, and access to a lipid reservoir, essential for the large increase in nuclear surface area during the cell cycle. We present experimental confirmation of the novel and non-trivial geometries predicted by our model, which has no free parameters. We also use the model to provide insight into the mechanical basis of previously described defects in nuclear division, including abnormal nuclear shapes and loss of nuclear envelope integrity. The model predicts that (i) despite differences in structure and composition, fission yeast nuclei and vesicles with fluid lipid bilayers have common mechanical properties; (ii) the S. pombe nucleus is not lined with any structure with shear resistance, comparable to the nuclear lamina of higher eukaryotes. We validate the model and its predictions by analyzing wild type cells in which ned1 gene overexpression causes elongation of an intranuclear microtubule bundle that deforms the nucleus of interphase cells

Spatial Frequency-Based Analysis of Mean Red Blood Cell Speed in Single Microvessels: Investigation of Microvascular Perfusion in Rat Cerebral Cortex

BACKGROUND: Our previous study has shown that prenatal exposure to X-ray irradiation causes cerebral hypo-perfusion during the postnatal development of central nervous system (CNS). However, the source of the hypo-perfusion and its impact on the CNS development remains unclear. The present study developed an automatic analysis method to determine the mean red blood cell (RBC) speed through single microvessels imaged with two-photon microscopy in the cerebral cortex of rats prenatally exposed to X-ray irradiation (1.5 Gy). METHODOLOGY/PRINCIPAL FINDINGS: We obtained a mean RBC speed (0.9±0.6 mm/sec) that ranged from 0.2 to 4.4 mm/sec from 121 vessels in the radiation-exposed rats, which was about 40% lower than that of normal rats that were not exposed. These results were then compared with the conventional method for monitoring microvascular perfusion using the arteriovenous transit time (AVTT) determined by tracking fluorescent markers. A significant increase in the AVTT was observed in the exposed rats (1.9±0.6 sec) as compared to the age-matched non-exposed rats (1.2±0.3 sec). The results indicate that parenchyma capillary blood velocity in the exposed rats was approximately 37% lower than in non-exposed rats. CONCLUSIONS/SIGNIFICANCE: The algorithm presented is simple and robust relative to monitoring individual RBC speeds, which is superior in terms of noise tolerance and computation time. The demonstrative results show that the method developed in this study for determining the mean RBC speed in the spatial frequency domain was consistent with the conventional transit time method

Fibro-Vascular Coupling in the Control of Cochlear Blood Flow

Transduction of sound in the cochlea is metabolically demanding. The lateral wall and hair cells are critically vulnerable to hypoxia, especially at high sound levels, and tight control over cochlear blood flow (CBF) is a physiological necessity. Yet despite the importance of CBF for hearing, consensus on what mechanisms are involved has not been obtained.We report on a local control mechanism for regulating inner ear blood flow involving fibrocyte signaling. Fibrocytes in the super-strial region are spatially distributed near pre-capillaries of the spiral ligament of the albino guinea pig cochlear lateral wall, as demonstrably shown in transmission electron microscope and confocal images. Immunohistochemical techniques reveal the inter-connected fibrocytes to be positive for Na+/K+ ATPase β1 and S100. The connected fibrocytes display more Ca(2+) signaling than other cells in the cochlear lateral wall as indicated by fluorescence of a Ca(2+) sensor, fluo-4. Elevation of Ca(2+) in fibrocytes, induced by photolytic uncaging of the divalent ion chelator o-nitrophenyl EGTA, results in propagation of a Ca(2+) signal to neighboring vascular cells and vasodilation in capillaries. Of more physiological significance, fibrocyte to vascular cell coupled signaling was found to mediate the sound stimulated increase in cochlear blood flow (CBF). Cyclooxygenase-1 (COX-1) was required for capillary dilation.The findings provide the first evidence that signaling between fibrocytes and vascular cells modulates CBF and is a key mechanism for meeting the cellular metabolic demand of increased sound activity