We develop a theoretical study which allows us to identify and quantify the relevant aspects that trigger the shape transition. The lipid bilayer tries to expand its outer leaflet in order to accommodate the excess area, whereas the cytoskeleton opposes resistance to this type of deformations, preserving more compact shapes. The subtle interplay between both membrane structures determines the equilibrium morphology of the cell. The cytoskeleton is fundamental to ensure the stability of the healthy shape, the discocyte, against changes in the membrane composition.
However, it is not severely stressed under weak deformations in which low curvatures are involved. Our results show that the energetic scale of these shape transitions is of hundreds of kbT, demonstrating the large stability of these shapes.
Based on the knowledge gained from the theoretical study we also analyze a series of experiments in which echinocytes are mechanically perturbed by a AFM tip, inducing shape transitions towards the healthy discocyte in a controlled manner. In the second Part, we derive a phase-field method for membrane modeling. Phase-field methods have been extensively used for the study of interface phenomena, though with few applications to membranes.
We present a new model which accounts for the membrane elasticity, and couples the membrane dynamics with an external fluid, whose hydrodynamics is dictated by the Navier-Stokes equation. We derive the expression of the stress tensor which allows us to recover the stress profile of the membrane. We also obtain the membrane equilibrium equations, proving that in the macroscopic limit our phase-field model recovers the correct expressions given by the elastic theory of membranes.
Effect of chronic kidney disease on red blood cell rheology
In the third Part we make use of this phase-field model to study the behaviour of RBCs in flow in narrow channels, of width similar to that of the cell. We consider pressure-driven flows as they relevant for both in vivo and in vitro circulation. We carry out simulations by means of a lattice-Bolztmann method. Our study highlights the crucial role of the RBC shape, softness and deformability to explain its complex behaviour and rheological properties. RBCs flowing at low concentratrions, when they do not interact with other cells and the dynamics is governed by the interaction with the cell, are shown to migrate lateral towards the wall, avoiding the axial position.
In this low to medium shear rate range, the cells wiggle with respect to the neighboring cells allowing flow. The influence of aggregation properties on the viscoelasticity diminish and the influence of red cell deformability begins to increase. As shear rates become large, red blood cells will stretch or deform and align with the flow.
Cell layers are formed, separated by plasma, and flow is now attributed to layers of cells sliding on layers of plasma. The cell layer allows for easier flow of blood and as such there is a reduced viscosity and reduced elasticity. The viscoelasticity of the blood is dominated by the deformability of the red blood cells.
Maxwell Model concerns Maxwell fluids or Maxwell material. Maxwell model is made to estimate local conservative values of viscoelasticity by a global measure in the integral volume of the model to be transposed to different flow situations. Blood is a complex material where different cells like red blood cells are discontinuous in plasma.
Their size and shape are irregular too because they are not perfect spheres. In theory, a fluid in a Maxwell Model behaves exactly similarly in any other flow geometry like pipes, rotating cells or in rest state. But in practice, blood properties vary with the geometry and blood has shown being an inadequate material to be studied as a fluid in common sense.
So Maxwell Model gives trends that have to be completed in real situation followed by Thurston model  in a vessel regarding distribution of cells in sheath and plug flows. If a small cubical volume of blood is considered, with forces being acted upon it by the heart pumping and shear forces from boundaries.
Influence of regular physical activity on blood rheology | European Heart Journal | Oxford Academic
The change in shape of the cube will have 2 components:. When the force is removed, the cube would recover partially. The elastic deformation is reversed but the slippage is not. This explains why the elastic portion is only noticeable in unsteady flow. In steady flow, the slippage will continue to increase and the measurements of non time varying force will neglect the contributions of the elasticity. Figure 1 can be used to calculate the following parameters necessary for the evaluation of blood when a force is exerted. A sinusoidal time varying flow is used to simulate the pulsation of a heart.
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A viscoelastic material will be somewhere in between 0 and 90 degrees. Similarly, the complex dynamic modulus G can be obtained by taking the ratio of the complex shear stress to the complex shear strain. Relating the equations to common viscoelastic terms we get the storage modulus, G', and the loss modulus, G". A viscoelastic Maxwell material model is commonly used to represent the viscoelastic properties of blood. It uses purely viscous damper and a purely elastic spring connected in series.
Analysis of this model gives the complex viscosity in terms of the dashpot constant and the spring constant. One of the most frequently used constitutive models for the viscoelasticity of blood is the Oldroyd-B model. There are several variations of the Oldroyd-B non-Newtonian model characterizing shear thinning behavior due to red blood cell aggregation and dispersion at low shear rate. Here we consider a three-dimensional Oldroyd-B model coupled with the momentum equation and the total stress tensor.
In the Oldroyd-B model, the relation between the shear stress tensor B and the orientation stress tensor A is given by:. S and B are defined as follows:. Red blood cells are subjected to intense mechanical stimulation from both blood flow and vessel walls, and their rheological properties are important to their effectiveness in performing their biological functions in the microcirculation. There are several methods used to explore the mechanical properties of red blood cells such as:.
These methods worked to characterize the deformability of the red blood cell in terms of the shear, bending, area expansion moduli, and relaxation times.
Other techniques have been implemented such as photoacoustic measurements. This technique uses a single-pulse laser beam to generate a photoacoustic signal in tissues and the decay time for the signal is measured. According to the theory of linear viscoelasticity, the decay time is equal to the viscosity-elasticity ratio and therefore the viscoelasticity characteristics of the red blood cells could be obtained. Another experimental technique used to evaluate viscoelasticity consisted of using Ferromagnetism beads bonded to a cells surface.
Forces are then applied to the magnetic bead using optical magnetic twisting cytometry which allowed researchers to explore the time dependent responses of red blood cells. Complex Dynamic modulus G can be used to represent the relations between the oscillating stress and strain:. From the above relations, the components of the complex modulus are determined from a loop that is created by comparing the change in torque with the change in time which forms a loop when represented graphically.
The hysteresis shown in figure 3 represents the viscoelasticity present in red blood cells. It is unclear if this is related to membrane molecular fluctuations or metabolic activity controlled by intracellular concentrations of ATP. Further research is needed to fully explore these interaction and to shed light on the underlying viscoelastic deformation characteristics of the red blood cells.
When looking at viscoelastic behavior of blood in vivo , it is necessary to also consider the effects of arteries , capillaries , and veins. The viscosity of blood has a primary influence on flow in the larger arteries, while the elasticity, which resides in the elastic deformability of red blood cells, has primary influence in the arterioles and the capillaries.
Arterial walls are anisotropic and heterogeneous, composed of layers with different bio-mechanical characteristics which makes understanding the mechanical influences that arteries contribute to blood flow very difficult. From a medical standpoint, the importance of studying the viscoelastic properties of blood becomes evident.
With the development of cardiovascular prosthetic devices such as heart valves and blood pumps, the understanding of pulsating blood flow in complex geometries is required. A few specific examples are the effects of viscoelasticity of blood and its implications for the testing of a pulsatile Blood Pumps. This has also led the way for developing a blood analog in order to study and test prosthetic devices.
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The classic analog of glycerin and water provides a good representation of viscosity and inertial effects but lacks the elastic properties of real blood. One such blood analog is an aqueous solution of Xanthan gum and glycerin developed to match both the viscous and elastic components of the complex viscosity of blood. Normal red blood cells are deformable but many conditions, such as sickle cell disease , reduce their elasticity which makes them less deformable.
Red blood cells with reduced deformability have increasing impedance to flow, leading to an increase in red blood cell aggregation and reduction in oxygen saturation which can lead to further complications. The presence of cells with diminished deformability, as is the case in sickle cell disease, tends to inhibit the formation of plasma layers and by measuring the viscoelasticity, the degree of inhibition can be quantified. In early theoretical work, blood was treated as a non-Newtonian viscous fluid. Initial studies had evaluated blood during steady flow and later, using oscillating flow.
Thurston, of the University of Texas, first presented the idea of blood being viscoelastic in The previous studies that looked at blood in steady flow showed negligible elastic properties because the elastic regime is stored in the blood during flow initiation and so its presence is hidden when a flow reaches steady state. The early studies used the properties found in steady flow to derive properties for unsteady flow situations. The relationships between shear stress and shear rate for blood must be determined experimentally and expressed by constitutive equations. Given the complex macro-rheological behavior of blood, it is not surprising that a single equation fails to completely describe the effects of various rheological variables e.
Thus, several approaches to defining these equations exist, with some the result of curve-fitting experimental data and others based on a particular rheological model. Heparin anticoagulation is commonly used to prevent blood clotting during surgeries and to treat thromboembolic diseases. One of the common adverse side- effects associated with heparin therapy is bleeding. Reversal of heparin anticoagulation activity is needed after surgical procedures such as the cardiopulmonary bypass to prevent excessive bleeding.
Currently, protamine is the only clinically available heparin antidote, however, associated with severe limitations and adverse effects. Protamine has a very narrow therapeutic window; excess protamine has an inherent anticoagulant effect. Protamine activates complement, induces fibrinolysis, causes lung-injury after heparin reversal, and is only partially effective against low molecular weight heparins LWMHs and other heparin derivatives such as fondaparinux. To circumvent the limitations of protamine, researchers are in active search for alternate antidotes with an improved safety profile that works against all the clinically used heparins.
The dynamics of blood clot formation during heparin neutralization could provide important information regarding the clot stabilization and effectiveness of the neutralization strategy.
Red Blood Cell mechanics: from membrane elasticity to blood rheology
This may also provide information regarding the potential of re-bleeding after heparin reversal. The rheological characterization of the blood clot formation with a higher sensitivity and additional measurement parameters could unravel such information to aid the current practice. Rheological characterization is one of the widely used methodologies to analyze the global static and dynamic behavior of the non-Newtonian, viscoelastic fluids, including blood and blood clot.
The advantage of rheometry using a standard rotational rheometer is that an adjustable oscillatory frequency under a controlled oscillatory amplitude within the linear viscoelastic range can be used. Providing an oscillatory shear within the linear viscoelastic range would be one of the critical control parameters since fibrin gel is known to experience strain hardening under nonlinear oscillation.