hemorheology hemorheology hemorheology, also spelled haemorheology (from greek ‘αἷμα, haima 'blood' and rheology, from greek ῥέω rhéō, 'flow' and -λoγία, -logia 'study of'), or blood rheology, is the study of flow properties of blood and its elements of plasma and cells. proper tissue perfusion can occur only when blood's rheological properties are within certain levels. alterations of these properties play significant roles in disease processes. blood viscosity is determined by plasma viscosity, hematocrit (volume fraction of red blood cell, which constitute 99.9% of the cellular elements) and mechanical properties of red blood cells. red blood cells have unique mechanical behavior, which can be discussed under the terms erythrocyte deformability and erythrocyte aggregation. because of that, blood behaves as a non-newtonian fluid. as such, the viscosity of blood varies with shear rate. blood becomes less viscous at high shear rates like those experienced with increased flow such as during exercise or in peak-systole. therefore, blood is a shear-thinning fluid. contrarily, blood viscosity increases when shear rate goes down with increased vessel diameters or with low flow, such as downstream from an obstruction or in diastole. blood viscosity also increases with increases in red cell aggregability. blood viscosity blood viscosity is a measure of the resistance of blood to flow. it can also be described as the thickness and stickiness of blood. this biophysical property makes it a critical determinant of friction against the vessel walls, the rate of venous return, the work required for the heart to pump blood, and how much oxygen is transported to tissues and organs. these functions of the cardiovascular system are directly related to vascular resistance, preload, afterload, and perfusion, respectively. the primary determinants of blood viscosity are hematocrit, red blood cell deformability, red blood cell aggregation, and plasma viscosity. plasma's viscosity is determined by water-content and macromolecular components, so these factors that affect blood viscosity are the plasma protein concentration and types of proteins in the plasma. nevertheless, hematocrit has the strongest impact on whole blood viscosity. one unit increase in hematocrit can cause up to a 4% increase in blood viscosity. this relationship becomes increasingly sensitive as hematocrit increases. when the hematocrit rises to 60 or 70%, which it often does in polycythemia, the blood viscosity can become as great as 10 times that of water, and its flow through blood vessels is greatly retarded because of increased resistance to flow. this will lead to decreased oxygen delivery. other factors influencing blood viscosity include temperature, where an increase in temperature results in a decrease in viscosity. this is particularly important in hypothermia, where an increase in blood viscosity will cause problems with blood circulation. clinical significance many conventional cardiovascular risk factors have been independently linked to whole blood viscosity. cardiovascular risk factors linked independently to whole blood viscosity hypertension total cholesterol vldl-cholesterol ldl-cholesterol hdl-cholesterol (negative correlation) triglycerides chylomicrons diabetes mellitus and insulin resistance metabolic syndrome obesity cigarette smoking male gender age anemia can reduce blood viscosity, which may lead to heart failure. furthermore, elevation of plasma viscosity correlates to the progression of coronary and peripheral artery diseases. normal level in pascal-seconds (pa·s), the viscosity of blood at 37 °c is normally 3 × 10−3 to 4 × 10−3, respectively 3 - 4 centipoise (cp) in the centimetre gram second system of units. μ = ( 3 ∼ 4 ) ⋅ 10 − 3 p a ⋅ s \,pa\cdot s} ν = μ ρ = ( 3 ∼ 4 ) ⋅ 10 − 3 1.06 ⋅ 10 3 = ( 2.8 ∼ 3.8 ) ⋅ 10 − 6 m 2 s ==(2.8\sim 3.8)\cdot 10^\, blood viscosity can be measured by viscometers capable of measurements at various shear rates, such as a rotational viscometer. blood viscoelasticity blood is a viscoelastic fluid, meaning that it possesses both viscous and fluid characteristics. the viscous component arises primarily through the viscosity of blood plasma, while the elastic component arises from deformation of the red blood cells. as the heart contracts, mechanical energy is transferred from the heart to the blood; a small part of the energy is dissipated by the viscosity of the suspension, another part is stored as elastic energy in the red blood cells, and the remaining energy is used to drive blood circulation and is thus converted to kinetic energy. viscoelastic fluids make up a larger class of fluids called non-newtonian fluids. the red blood cells occupy about half of the volume of blood and possess elastic properties. this elastic property is the largest contributing factor to the viscoelastic behavior of blood. the large volume percentage of red blood cells at a normal hematocrit level leaves little room for cell motion and deformation without interacting with a neighboring cell. calculations have shown that the maximum volume percentage of red blood cells without deformation is 58% which is in the range of normally occurring levels. due to the limited space between red blood cells, it is obvious that in order for blood to flow, significant cell to cell interaction will play a key role. this interaction and tendency for cells to aggregate is a major contributor to the viscoelastic behavior of blood. red blood cell deformation and aggregation is also coupled with flow-induced changes in the arrangement and orientation as a third major factor in its viscoelastic behavior. other factors contributing to the viscoelastic properties of blood is the plasma viscosity, plasma composition, temperature, and the rate of flow or shear rate. together, these factors make human blood viscoelastic, non-newtonian, and thixotropic. when the red cells are at rest or at very small shear rates, they tend to aggregate and stack together in an energetically favorable manner. the attraction is attributed to charged groups on the surface of cells and to the presence of fibrinogen and globulins. this aggregated configuration is an arrangement of cells with the least amount of deformation. with very low shear rates, the viscoelastic property of blood is dominated by the aggregation and cell deformability is relatively insignificant. as the shear rate increases the size of the aggregates begins to decrease. with a further increase in shear rate, the cells will rearrange and orient to provide channels for the plasma to pass through and for the cells to slide. 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 maxwell model concerns maxwell fluids or maxwell material. the material in maxwell model is a fluid which means it respects continuity properties for conservative equations : fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids. 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. complicating moreover blood volume shape, red cells are not identically distributed in a blood sample volume because they migrate with velocity gradients in direction to the highest speed areas calling the famous representation of the fåhræus–lindqvist effect, aggregate or separate in sheath or plug flows described by thurston. typically, the maxwell model described below is uniformly considering the material (uniform blue color) as a perfect distributed particles fluid everywhere in the volume (in blue) but thurston reveals that packs of red cells, plugs, are more present in the high speed region, if y is the height direction in the maxwell model figure, (y~h) and there is a free cells layer in the lower speed area (y~0) what means the plasma fluid phase that deforms under maxwell model is strained following inner linings that completely escape from the analytical model by maxwell. 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. the change in shape of the cube will have 2 components: elastic deformation which is recoverable and is stored in the structure of the blood. slippage which is associated with a continuous input of viscous energy. 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 - displacement due to elastic and viscous effects figure 1 can be used to calculate the following parameters necessary for the evaluation of blood when a force is exerted. shear stress: τ = f a shear strain: γ = d h shear rate: γ ˙ = v h = a sinusoidal time varying flow is used to simulate the pulsation of a heart. a viscoelastic material subjected to a time varying flow will result in a phase variation between τ and γ represented by ϕ . if ϕ = 0 , the material is a purely elastic because the stress and strain are in phase, so that the response of one caused by the other is immediate. if ϕ = 90°, the material is a purely viscous because strain lags behind stress by 90 degrees. a viscoelastic material will be somewhere in between 0 and 90 degrees. the sinusoidal time variation is proportional to e i ω t . therefore, the size and phase relation between the stress, strain, and shear rate are described using this relationship and a radian frequency, ω = 2 π f were f is the frequency in hertz. shear stress: τ ∗ = τ e − i ϕ =\tau e^ shear strain: γ ∗ = γ e − i π 2 =\gamma e^ shear rate: γ ˙ ∗ = γ ˙ e − i 0 ^=e^ the components of the complex shear stress can be written as: τ ∗ = τ ′ − i τ ″ =\tau '-i\tau ''} where τ ′ is the viscous stress and τ ″ is the elastic stress. the complex coefficient of viscosity η ∗ can be found by taking the ratio of the complex shear stress and the complex shear rate: η ∗ = τ ∗ γ ˙ ∗ = ( τ ′ γ ˙ + i τ ″ γ ˙ ) = η ′ + i η ″ =^=(+i)=\eta '+i\eta ''} similarly, the complex dynamic modulus g can be obtained by taking the ratio of the complex shear stress to the complex shear strain. g = τ ∗ γ ∗ = ( τ ″ γ + i τ ′ γ ) =(+i)} relating the equations to common viscoelastic terms we get the storage modulus, g', and the loss modulus, g". figure 2 - schematic of maxwell model using one dash-pot and one spring connected in series g = g ′ + i 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. η ∗ = η d