White Blood Cells: Morphology and Rheology as Related to Function
Understanding of blood flow is essential for the elucidation of cardiovascular disease, the leading causes of death worldwide. This paper presents an overview of the laws of the rheological behavior of blood. Blood is a complex fluid in perpetual renewal, which provides a number of essential functions for maintaining life. Suspended particles are red blood cells the majority , white blood cells, platelets, and other substances such as proteins, vitamins etc. The suspending liquid is plasma, aqueous solution of electrolytes and organic substances, mostly proteins.
Blood flows continuously into the vasculature of the body, under the impulse of the heart beat. Since the discovery of the large circulation attributed by historians to William Harvey in the small pulmonary circulation is higher in Harvey , the Blood research has continued, bringing together several scientific specialties anatomy, medicine, physiology, rheology…. From the standpoint of the rheology, the blood is considered as a concentrated colloidal suspension of erythrocytes.
The relationship between shear rate and apparent viscosity of a suspension of erythrocytes in plasma was studied by Shu Dog . The interpretation of this rheological behavior is based on two processes:. These two processes making the viscosity depending on the flow rate, are closely related to the properties of plasma and red blood cells.
In addition to the applied shear rate, the apparent viscosity of blood depends on other important factors which are: the volume concentration of red blood cells hematocrit or H t , mechanical properties and plasma viscosity itself a function of the fibrinogen concentration and albumin. The varied geometry of the circulatory system arteries, veins, capillaries… gives rise to various flow regimes, themselves responsible for the structuring of erythrocytes according to the strength of the shear. The blood viscosity depends on suspension components plasma, particles but also the diameter of the vessel and the walls of its deformability.
Thus the red blood cell stacks are important at the large vessels and induce an increase in viscosity. Red blood cells have the ability to individually align, and pass through the capillary whose diameter is less than the average diameter cells. The viscosity of blood is generally normalized by plasma viscosity, and is then called relative viscosity.
A mathematical formulation of the viscosity should consider these parameters to be appropriate. The rheological models empirical or semi empirical that we will study are able to adjust the results of experimental measurements over a wide range of shear rate with the same parameters. To represent the dynamic viscosity of blood depending on the shear rate, we cite four models very present in the literature  . It should be noted that these models already exist for non-Newtonian fluids in particular rheo-fluids but were either adapted or developed for blood. Each model contains parameters identified as important to describe the rheological behavior of blood.
We cite as examples of parameters: the yield stress, the Newtonian limiting viscosity and the molecular composition of the blood. The relation or power model is valid only on a limited range of shear rates without low and high shear rates. This led to a model with three parameters, proposed by Cross  and extending the power-law model to a wider range of shear rates:. This power-law model with two parameters has been proposed to coincide with the experimental curve obtained from a blood sample for which hematocrit and chemical composition were known.
There is a statistical correlation between the three most influential parameters on the fluid shear rate, hematocrit, proteins concentration in plasma . The apparent viscosity is:. If the shear rate is very low , the fluid is Newtonian but if the shear rate is strong , the fluid follows a power law. Suitable for blood, the Carreau model has several variants as the model of Carreau-Gambaruto  -  which has the parameters below:. This model was developed by Quemada  to represent the apparent viscosity of the blood based on the shear rate and taking into account the hematocrit.
If in the literature, many models are described or used, two models are cited as being more appropriate to describe the measured viscosity: Quemada model and Carreau-Gambaruto model. Figure 1 shows a comparison between the two models and the measured viscosity . The experimental values correspond to a blood of non-smoking healthy man of 56 years. Overall, the curve of the viscosity can be divided into three areas corresponding respectively to the low, the means and high shear rates.
The term viscoelasticity is used for blood for the first time by GB Thurston  who explains that for this region, the viscoelastic properties are due to aggregation and not to deformability which is negligible. For medium as , the internal stress is sufficient to break the aggregates. With shear rate increasing, the disaggregated cells are gradually moving in the flow direction. In this region, the aggregation influence on the viscoelasticity decreases in favor of the deformability influence. The point of inflection between region 1 and region 2 is well explained by the phenomenon of relaxation of the blood break of RG rouleaux.
When the shear rate reaches high levels, normal RG stretches or deforms and align with the flow.
White Blood Cells by U. Bagge, Gustav V. R. Born | Waterstones
The blood forms RG layers stretched and sliding on plasma layers. In terms of energy, viscosity is related to the dissipated energy in the flow due to the deformation and slip of. Figure 1. Comparison between measured viscosity  , Carreau-Gambaruto-model and Quemada model. The arrow shows the inflection point on the viscosity curve. Elasticity is due to the stored energy in the flow direction due to the deformation and the red blood cells.
Very recently, Brust et al. The study of the blood rheological behavior is very important for understanding blood flow, which helps to detect and consequently to treat cardiovascular disease. Existing models of viscosity including models Carreau- Gambaruto and Quemada are approximate because it is difficult to introduce all the factors affecting the viscosity. Viscoelasticity of blood and the newly established viscoelasticity of plasma should bring more light on blood rheology.
The discontinuity inflection point on the experimental curve presents a notable difference with the models. We suggest that the models of the viscosity must consider the relaxation of the system as it is mentioned in the work of G. Thurston . We also note that this study is based on one experimental measure of blood viscosity. We will work in the future on an averaged experimental viscosity. Science, , Chemical Engineering Science, , The shear modulus decreased as temperature increased from 5 to 45C Waugh and Evans, Malaria invasion cause significant increases in shear moduli values Mills, Diez-Silva et al.
Bending modulus. Bending modulus or fluxural modulus B of a membrane is determined by the energy needed to deform a membrane from its original curvature to some other curvature. The bending modulus B of a 2-D membrane is described as. C1 and C2 are two principle curvatures, and C3 is the curvature in the stress-free state Helfrich, ; Evans, Bending of a 2-D structure involves both area compression and expansion. The elastic bending moduli B of lipid bilayer is determined by chemical compositions of the lipids, and there is a broad range of reported bending moduli for lipid bilayers Boal, The elastic bending moduli B of RBC membranes have been measured with various techniques.
The bending modulus B of RBCs does not significantly change with temperature Nash and Meiselman, or cell Hb concentration for both normal and sickle cells Evans, Mohandas et al. The recent experiments Betz, Lenz et al. However, several other techniques have measured lower bending moduli of RBC membranes.
White Blood Cells: Morphology and Rheology as Related to Function (Microcirculation Review)
While the elasticic property of RBC membranes characterizes its resistance to deformation, the viscous property characterized its resistance to a rate of deformation Hochmuth and Waugh, The viscous properties of RBC membranes can be determined by 3-D cytoplasmic viscosity and 2-D membrane viscosity. Cytoplasmic viscosity. Cytosolic viscosity depends on the concentration and viscosity of Hb. By measuring the dynamic contour fluctuations of RBC membrane, the cytoplasmic viscosity has been obtained in the range of mPa s Yoon, Hong et al.
Recently, the dynamic membrane fluctuation measurements retrieved the cytoplasmic visocity of the RBCs at physiological osmotic pressure as mPa s, and the cytosol viscosity increases monotonically from with increasing osmolality Park, Best et al. Membrane viscosity. The major source of viscous dissipation in RBC membranes is the membrane viscosity. During the recovery process after large deformation of RBCs, 2-D membrane viscosity dominates energy dissipation Evans and Hochmuth, Considering viscous dissipation due to a 2-D membrane viscosity, the modified version of the shear force from Eq.
The 2-D surface viscosity of RBC membranes has been measured by several experiments. Using this method, the 2-D membrane viscosity values of RBC membranes have been reported in the range of 0.
Using mathematical models, the mechanics of the membrane cortex structures has been simulated. Using a worm-like-chain model with surface and bending energy, the force-displancement relations for the spectrin network of RBCs have been described Discher, Boal et al. The viscoelastic properties of the RBC membrane was described using an effective continuum membrane model that simulates a finite-thickess 2-D continuum plane model with in-plane shear modulus and bending modulus Dao, Lim et al. Recently developed numerical models accurately describes the complex viscoelastic properties of RBCs deformabilty Fedosov, Caswell et al.
The RBC deformability can influence blood microcirculation since viscosity and flow can be significantly changed by the viscoelastic properties of RBCs. Viscosity of liquid characterizes its resistance to flow under certain deforming force, especially shear stress. Under laminar flow conditions where particles move parallel to adjacent neighbors with minimal turbulence, the fluidity is classified by the dependence of viscosity to shear strain or shear stress: 1 Newtonian fluid, if the viscosity is independent of shear stress or shear strain so that shear stress is linearly proportional to shear strain, 2 non-Newtonian fluid whose viscosity either decrease shear-thinning or increase shear-thickening depending on the changes of shear stress Merrill, Blood is non-Newtonian fluid which exhibits shear-thinning behavior.
Blood viscosity decreases at high shear stress due to the deformation of RBCs, while it increases at low shear stress because RBCs aggregate with each others and form stacked coin structure, called rouleaux Shiga, Maeda et al. Whole blood is a two-phase liquid consisting of a liquid medium plasma and formed elements such as RBCs, white blood cells, and platelets. Modified, with permission, from Somer and Meiselman,; Baskurt, Plasma is a Newtonian fluid which viscosity in normal condition varies 1.
Elevated levels of fibrinogen concentration in plasma enhance RBC aggregation and thus it increases blood viscosity. Formed elements in the stream lines of laminar flow of blood can be considered as the source of turbulence which significantly increases blood viscosity. Among formed elements, RBCs cause the most significant effects since RBCs concentration is the highest among the formed elements in blood. The blood viscosity increases as hematocrit increases; the hematocrit effect becomes more severe when shear stress decreases since more aggregation of RBCs takes place Dormandy, ; Baskurt, Microcirculation transports blood to the small vessels in the vasculature embedded within organs.
Flow dynamics in microcirculation requires deep consideration of 1 fluid dynamics in capillaries, 2 interaction between formed elements with vessel walls, and 3 the structure and network of microvessels. Blood flow in microcirculation is not only determined by the geometric features of blood vessels and hydrostatic blood pressure, but also affected by the rheological properties. RBC deformability can significantly alter blood flow in microcirculation Chien, The reduction in RBC deformability under certain physiological or pathological conditions results into the retardation of blood-flow thourgh the microcirculation, which plays important roles in the stages of peripheral vascular insufficiency Reid, Dormandy et al.
Micropipette aspiration techniques have been extensively used to measure the mechanical properties of RBC membranes Evans and La Celle, ; Shiga, Maeda et al. When negative pressure is applied, RBC membrane is aspirated into the micropipette and the amount of aspiration depends on the viscoelastic properties of cell membrane. Detailed measurement techniques vary depending on the mechanical property of interest Fig. The area expansion modulus of RBC membranes can be measured by using micropipette aspiration based on Eq.
In order to measure the shear modulus of RBC membranes, the second method Fig. Various methods for micropipette aspiration. A Measuring pressure P to aspirate the distance same with the micropipette radius. B Measuring the ratio between the aspirated length of membrane D and the micropipette radius at a certain negative pressure. Reproduced, with permission, from Evans and La Celle, The value for B depends directly on the magitude of the aspiration pressure when RBCs start to buckle and inversely on the pipette area; measuring negative pressure with varying radius of the pipette can measure B of RBCs.
The measured value for B was By measuring the time for recovering original shape from releasing negative pressure, the 2-D viscosity of RBC membranes can also be obtained by Eq. Atomic force microscopy AFM is a tip-scanning technique that images topographies of materials in atomic or molecular scale Binnig, Quate et al.
It uses a cantilever with a sharp tip as a probe. Depending on the amount of force to apply or sensitivity, diverse tip shapes are used such as triangular, parabolic, or cylindrical shapes Weisenhorn, Khorsandi et al. As a tip scans over a sample with physical contact, the vertical motion of the tip is monitored by photodiodes which precisely detect small changes in laser beam position reflected from the tip.
As shown in Figs. A Topogram of normal RBCs. B Detailed texture of the RBC membrane surface. C Indentation depth measurement. D Different force-versus-indentation depth curves of RBCs in various conditions: a. G6PD deficiency; and n. Reproduced, with permission, from Kamruzzahan, Kienberger et al.
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The displacement of the stage required for the same deflection of the tip is different between solid- and soft-materials, from which applied forces can be calibrated. The Poisson ratio is 0. RBCs from hereditary spherocytosis, thalassemia Dulinska, Targosz et al. Light refraction at a sample induces linear momentum change, resulting into trapping forces Fig. High numerical aperture NA objective lens is used to generate a tightly focused optical trap, and its trapping force is governed by the refractive indices of sample and surrounding medium, laser power, and sample size; optical force to trap particles much smaller than laser wavelength can be described by Rayleigh scattering theory, while trapping samples much larger than laser wavelength belongs to Mie scattering regime Ashkin, Dziedzic et al.
Principles of optical tweezers. A Laser beam with gradual intensity transfers linear momentum to a microsphere to escape from the beam center. B Focused Gaussian beam exerts trapping force. Modified, with permission, from Svoboda and Block,; Henon, Lenormand et al. Since optical tweezers can apply forces at the pN level, it has been employed for measuring the deformability of RBCs.
Measurements of the mechanical properties of RBCs with optical tweezers can be done either by applying optical force to microspheres attached to RBCs Henon, Lenormand et al. The change in the projected diameter of the RBC in response of optical force is converted to shear modulus of the RBC using mathematical membrane models. Optical stretcher, a variant of optical tweezers, uses two diverging laser beams from opposite directions Guck, Ananthakrishnan et al. Linear momentum changes by two laser beams apply stretching force to the RBC along the optical axis, and the RBC deformations under varying optical force are measured from which mechnical properties are retrieved.
Optical tweezers can also be used for detecting membrane fluctuation dynamics of RBCs by imposing a deformation Yoon, Kotar et al. Magnetic twisting cytometry MTC applies both static and oscillating magnetic field to ferromagnetic microbeads attached to the surface of cell membrane Wang, Butler et al. Depending on the applied magnetic field, the microbeads exhibit both translational and rotational motion, which applies torques to the cell membrane. By varying oscillating frequency 0. Reproduced, with permission, from Puig-de-Morales-Marinkovic, Turner et al.
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Quantitaitve phase imaging technqiues measure the electric field, i. Employing the principle of laser interference, electric field information of target sample is modulated onto intereferograms recorded by a CCD camera. By using appropriate field retrieval algorithms, the field information can be retrieved from the measured holograms Debnath and Park, Typical interferogram and quantitative phase image of a RBC are shown in Fig.
Quantitative phase imaging techniques can measuring dynamic membrane fluctuations of RBCs Popescu, Ikeda et al. Dynamic membrane fluctuation, consisting of submicron displacement of the membrane, has a strong correlation with deformability of RBCs and can be altered by biochemical changes in protein level Waugh and Evans, Quantitative phase imaging. A Schematic of the principle of quantitative phase imaging. Reproduced, with permission, from Park, Best et al. Diffraction phase microscopy DPM , a highly stable technique for quantitative phase imaging, has been widely used for investigating the deformability of RBCs.
Employing common-path laser interferometry, DPM provides full-field quantitative phase imaging with unprecedented stability Park, Popescu et al. DPM measured spatiotemporal coherency in dynamic membrane fluctuations Popescu, Park et al. Recently, integrated with a mathematical model, DPM provide the mechanical properties of individual RBCs from membrane fluctuations: shear modulus, bending modulus, area expansion modulus, and cytoplasmic viscosity Park, Best et al.
Employing spectroscopic quantitative phase imaging, cytoplasmic Hb concentration that is tightly related to the cytoplasmic viscosity, can also be simultaneously quantified Park, Yamauchi et al. In addition, polarization-sensitive quantitative phase microscopy will be potentially used for the study of sickle cell disease and its implications to RBC deformability Kim, Jeong et al.
Although dynamic light scattering have been extensively used in combination with ektacytometry, it provides averaged signals from many RBCs. Thus it is difficult to access the deformability of individual RBCs.
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Fourier transform light scattering FTLS provides both static and dynamic light scattering signal from individual cells. Light field, measured by quantitative phase microscopy or digital holographic microscopy, contains both amplitude and phase information, and thus far-field light scattering pattterns can be directly calculated by numerically propagating the measured field — technically applying Fourier transformation Ding, Wang et al. FTLS technique can provide both morphological and rheological information about individual biological cells. Due to its capability of measuring light scattering signals from individual cells with high signal-to-noise ratio, FTLS has been employed to study several pathophysiological effects to the deformabiltiy of RBCs, including malaria infection Park, Diez-Silva et al.
Blood viscometer measures the viscosity of blood over a wide range of shear rates. Blood viscometer controls either shear stress or shear rate of blood using rational objects. Stress-controlled blood viscometer applys a constant torque which corresponds to constant rotational speed in a well-designed rotational rheometer. In a rate-controlled system, applied torque is controlled by a stress-sensing device so that a constant rotational speed is achieved.
Viscometers can be classified by the cylinder shape: a concentric cylinder, a cone plate, and a parallel plate viscometer Fig. Cylinder-type viscometer uses two concentric cylinders: a rotational inner cup and a stationary outer cylinder. Time-independent shear rate can be precisely measured by concentric cylinder viscometer Nguyen and Boger, Parallel plate viscometer is a simplified version of the cone and plate viscometer and has a advantage of flexible space between two parallel plates. The viscous fluid can confined and rotated in narrow space between two circular parallel plates Gent, Schematic diagrams of typical viscometers.
Ektacytometer employes a laser diffraction technique with blood viscometer in order to measure RBC deformabiltiy. Conventional blood viscometer applys controlled shear stress to the RBCs in the blood viscometer, and deformability of RBCs can be measured from laser diffraction pattern. Ektacytometer consists of a concentric rotational outer cup and a stationaly inner cylinder; outer cup produces varying shear stress field on blood Fig. Through the measurement of diffraction patterns of the laser passing through the blood, RBC deformability can be obtained.
The RBC deformation is quantitatively calculated from the scattered laser beam intensity pattern. Under a certain shear rate, isointensity curves in the intensity pattern of the scattered beam will show elliptical shapes, which represent elliptically deformed RBC population Bessis, Mohandas et al. DI values are measured at different angular velocities and thus different shear rate of the outer cyliner in the ektacytometer. Ektacytometer is a simple and effective technique to measure the deformability of RBC population, and it has been widely used for the study pathophysiology of RBCs.
Abnormal deformability in RBCs from patients with hereditary pyropoikilocytosis, hereditary spherocytosis, and Hb CC disease were studies by ektacytometer Mohandas, Clark et al. Microfludic device reduced space, labor, and measurement time on numerous experiments, and also enabled precise control and manipulation of the small volume of samples. Microfludic device has been used to study the deformabiltiy of RBCs. Microfluidic channel with a few micrometer diameter mimics micro-capillary structure in blood circulation system. Microfludic device was used to induce large deformation of RBCs and its mechanical behavior was studied Fig.
Reproduced, with permission, from Li, Lykotrafitis et al. For the study of sickle cell disease, microfluidic device has been used to measure the resistance change rate of blood flow under the sudden change of oxygen concentration Wang, Ding et al. Recently, microfludic channels with obstracles have measured the deformabiltiy of malaria infected RBCs in high throughput Bow, Pivkin et al. By applying a negative pressure, whole blood is subject to pass through holes in the membrane filter. The deformability of RBCs can affect the speed of flow. Since the filteration test requires for a relatively simple instrument and provides clinically relevant results with high reproducibility, it has been widely used in various studies related to RBC deformability, including the effects of diabetes Juhan, Buonocore et al.
This essential deformability is in turn affected by various physiological and pathological cues. Temperature plays important roles in RBC deformabilty. The elastic properties of RBC membrane were investigated as function of temperature using the micropipette aspiration technique Waugh and Evans, Due to the structual transitions of proteins occuring at certain critical tempertures, RBC deformabiltiy exhibits complex behaviors. Body temperature or febril temperature are particularly important in various pathophysiology of RBCs.