a s h 1 + i ω ( η d a s h e s p r i n g ) = η ′ − i η ″ =)}}=\eta '-i\eta ''} oldroyd-b model 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. a non newtonian flow is used which insures that the viscosity of blood μ ( h , d ) is a function of vessel diameter d and hematocrit h. in the oldroyd-b model, the relation between the shear stress tensor b and the orientation stress tensor a is given by: s + γ [ d s d t − δ v ⋅ s − s ⋅ ( δ v ) t ] = μ ( h , d ) [ b + γ ( d b d t − δ v ⋅ b − b ⋅ ( δ v ) t ) ] − g a + c 1 ( g a − c 2 i μ ( h , d ) 2 ) \left(ga-i}\right)} where d/dt is the material derivative, v is the velocity of the fluid, c1, c2, g, γ are constants. s and b are defined as follows: s = μ b + g a b = δ v + ( δ v ) t viscoelasticity of red blood cells figure 2 - schematic of maxwell model using one dash-pot and one spring connected in series 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. red blood cells by themselves have been shown to exhibit viscoelastic properties. there are several methods used to explore the mechanical properties of red blood cells such as: micropipette aspiration micro indentation optical tweezers high frequency electrical deformation tests these methods worked to characterize the deformability of the red blood cell in terms of the shear, bending, area expansion moduli, and relaxation times. however, they were not able to explore the viscoelastic properties. 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. t s ( t ) (t)} is the mechanical torque per unit bead volume (units of stress) and is given by: t s ( t ) = c h cos ⁡ θ (t)=ch\cos \theta } where h is the applied magnetic twisting field, θ is the angle of the bead’s magnetic moment relative to the original magnetization direction, and c is the bead constant which is found by experiments conducted by placing the bead in a fluid of known viscosity and applying a twisting field. complex dynamic modulus g can be used to represent the relations between the oscillating stress and strain: g = g ′ + i g ″ where g ′ is the storage modulus and g ″ is the loss modulus: g ′ = σ 0 ε 0 cos ⁡ ϕ \cos \phi } g ″ = σ 0 ε 0 sin ⁡ ϕ \sin \phi } where σ 0 and ε 0 are the amplitudes of stress and strain and ϕ is the phase shift between them. figure 3 - torque vs. displacement graph showing viscoelastic behavior 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 limits of t s ( t ) (t)} - d(t) loop and the area, a, bounded by the t s ( t ) (t)} - d(t) loop, which represents the energy dissipation per cycle, are used in the calculations. the phase angle ϕ , storage modulus g', and loss modulus g then become: ϕ = sin − 1 ⁡ 4 a π δ t s δ d \delta d}}} g ′ = δ t s δ d cos ⁡ ϕ \cos \phi } g ″ = δ t s δ d sin ⁡ ϕ = 4 a π ω δ d 2 \sin \phi = where d is the displacement. 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. effects of blood vessels 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. understanding wave propagation in arterial walls, local hemodynamics, and wall shear stress gradient is important in understanding the mechanisms of cardiovascular function. 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. medical reasons for a better understanding 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. strong correlations between blood viscoelasticity and regional and global cerebral blood flow during cardiopulmonary bypass have been documented. this has also led the way for developing a blood analog in order to study and test prosthetic devices. 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. history 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. professor george b. thurston, of the university of texas, first presented the idea of blood being viscoelastic in 1972. 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. advancements in medical procedures and devices required a better understanding of the mechanical properties of blood. constitutive equations 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.g., hematocrit, shear rate). 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. newtonian fluid model where has a constant viscosity at all shear rates. this approach is valid for high shear rates ( γ ˙ > 700 s − 1 >700\,s^ ) where the vessel diameter is much bigger than the blood cells. bingham fluid model takes into account the aggregation of red blood cells at low shear rates. therefore, it acts as an elastic solid under threshold level of shear stress, known as yield stress. einstein model where η0 is the suspending fluid newtonian viscosity, "k" is a constant dependent on particle shape, and h is the volume fraction of the suspension occupied by particles. this equation is applicable for suspensions having a low volume fraction of particles. einstein showed k=2.5 for spherical particles. μ a = μ 0 × ( 1 + k h ) =\times casson model where "a" and "b" are constants; at very low shear rates, b is the yield shear stress. however, for blood, the experimental data can not be fit over all shear rates with only one set of constants "a" and "b", whereas fairly good fit is possible by applying the equation over several shear rate ranges and thereby obtaining several sets of constants. τ 0.5 = a | γ | 0.5 + b 0.5 ^=^+b^ quemada model where k0, k∞ and γc are constants. this equation accurately fits blood data over a very wide range of shear rates. μ a = μ 0 ( 1 − 0.5 k h ) − 2 =^ k = k 0 + k inf γ 0.5 r 1 + γ 0.5 r +k__ \over _ γ r = γ γ c = \over other characteristics the fåhraeus effect main article: fåhræus effect the finding that, for blood flowing steadily in tubes with diameters of less than 300 micrometres, the average hematocrit of the blood in the tube is less than the hematocrit of the blood in the reservoir feeding the tube is known as the fåhræus effect. this effect is generated in the concentration entrance length of the tube, in which erythrocytes move towards the central region of the tube as they flow downstream. this entrance length is estimated to be about the distance that the blood travels in a quarter of a second for blood where red blood cell aggregation is negligible and the vessel diameter is greater than about 20 micrometres. the fåhræus–lindqvist effect main article: fåhræus–lindqvist effect as the characteristic dimension of a flow channel approaches the size of the particles in a suspension; one should expect that the simple continuum model of the suspension will fail to be applicable. often, this limit of the applicability of the continuum model begins to manifest itself at characteristic channel dimensions that are about 30 times the particle diameter: in the case of blood with a characteristic rbc dimension of 8 μm, an apparent failure occurs at about 300 micrometres. this was demonstrated by fåhraeus and lindqvist, who found that the apparent viscosity of blood was a function of tube diameter for diameters of 300 micrometres and less when they flowed constant-hematocrit blood from a well-stirred reservoir through a tube. the finding that for small tubes with diameters below about 300 micrometres and for faster flow rates which do not allow appreciable erythrocyte aggregation, the effective viscosity of the blood depends on tube diameter is known as the fåhræus–lindqvist effect.