Hemorheology

Many conventional cardiovascular risk factors have been independently linked to whole blood viscosity.

Anemia can reduce blood viscosity, which may lead to heart failure.[7] Furthermore, elevation of plasma viscosity correlates to the progression of coronary and peripheral artery diseases.[3][4]

In pascal-seconds (Pa·s), the viscosity of blood at 37 °C is normally 3 × 10⁻³ to 4 × 10⁻³,[8] respectively 3 - 4 centipoise (cP) in the centimetre gram second system of units.

${\displaystyle \mu =(3\sim 4)\cdot 10^{-3}\,Pa\cdot s}$

${\displaystyle \nu ={\frac {\mu }{\rho }}={\frac {(3\sim 4)\cdot 10^{-3}}{1.06\cdot 10^{3}}}=(2.8\sim 3.8)\cdot 10^{-6}\,{\frac {m^{2}}{s}}}$

Blood viscosity can be measured by viscometers capable of measurements at various shear rates, such as a rotational viscometer.[9]

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.[15] 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 [15] 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: ${\displaystyle \tau ={\frac {F}{A}}}$

Shear Strain: ${\displaystyle \gamma ={\frac {D}{H}}}$

Shear Rate: ${\displaystyle {\dot {\gamma }}={\frac {V}{H}}}$

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 ${\displaystyle \tau }$ and ${\displaystyle \gamma }$ represented by ${\displaystyle \phi }$. If ${\displaystyle \phi =0}$, 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 ${\displaystyle \phi }$ = 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 ${\displaystyle e^{i\omega t}}$. Therefore, the size and phase relation between the stress, strain, and shear rate are described using this relationship and a radian frequency, ${\displaystyle \omega =2\pi f}$ were ${\displaystyle f}$ is the frequency in Hertz.

Shear Stress: ${\displaystyle \tau ^{*}=\tau e^{-i\phi }}$

Shear Strain: ${\displaystyle \gamma ^{*}=\gamma e^{-i{\frac {\pi }{2}}}}$

Shear Rate: ${\displaystyle {\dot {\gamma }}^{*}={\dot {\gamma }}e^{-i0}}$

The components of the complex shear stress can be written as:

${\displaystyle \tau ^{*}=\tau '-i\tau ''}$

Where ${\displaystyle \tau '}$ is the viscous stress and ${\displaystyle \tau ''}$ is the elastic stress. The complex coefficient of viscosity ${\displaystyle \eta ^{*}}$ can be found by taking the ratio of the complex shear stress and the complex shear rate:[16]

${\displaystyle \eta ^{*}={\frac {\tau ^{*}}{{\dot {\gamma }}^{*}}}=({\frac {\tau '}{\dot {\gamma }}}+i{\frac {\tau ''}{\dot {\gamma }}})=\eta '+i\eta ''}$

Similarly, the complex dynamic modulus G can be obtained by taking the ratio of the complex shear stress to the complex shear strain.

${\displaystyle G={\frac {\tau ^{*}}{\gamma ^{*}}}=({\frac {\tau ''}{\gamma }}+i{\frac {\tau '}{\gamma }})}$

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

${\displaystyle G=G'+iG''}$

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.

${\displaystyle \eta ^{*}={\frac {\eta _{dash}}{1+i\omega ({\frac {\eta _{dash}}{E_{spring}}})}}=\eta '-i\eta ''}$

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.[17] A non Newtonian flow is used which insures that the viscosity of blood ${\displaystyle \mu (h,d)}$ 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:

${\displaystyle S+\gamma \left[{\frac {DS}{Dt}}-\Delta V\cdot S-S\cdot {(\Delta V)}^{T}\right]=\mu (h,d)\left[B+\gamma \left({\frac {DB}{Dt}}-\Delta V\cdot B-B\cdot {(\Delta V)}^{T}\right)\right]-gA+C_{1}\left(gA-{\frac {C_{2}I}{\mu (h,d)^{2}}}\right)}$

where D/Dt is the material derivative, V is the velocity of the fluid, C1, C2, g, ${\displaystyle \gamma }$ are constants. S and B are defined as follows:

${\displaystyle S=\mu B+gA}$

${\displaystyle B=\Delta V+(\Delta V)^{T}}$

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.[18] 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[19]
• 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.[20] 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.[21]

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.[22]

${\displaystyle T_{s}(t)}$ is the mechanical torque per unit bead volume (units of stress) and is given by:

${\displaystyle T_{s}(t)=cH\cos \theta }$

where H is the applied magnetic twisting field, ${\displaystyle {\theta }}$ 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:

${\displaystyle G=G'+iG''}$

where ${\displaystyle G'}$ is the storage modulus and ${\displaystyle G''}$ is the loss modulus:

${\displaystyle G'={\frac {\sigma _{0}}{\varepsilon _{0}}}\cos \phi }$

${\displaystyle G''={\frac {\sigma _{0}}{\varepsilon _{0}}}\sin \phi }$

where ${\displaystyle \sigma _{0}}$ and ${\displaystyle \varepsilon _{0}}$ are the amplitudes of stress and strain and ${\displaystyle \phi }$ 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 ${\displaystyle T_{s}(t)}$ - d(t) loop and the area, A, bounded by the ${\displaystyle T_{s}(t)}$ - d(t) loop, which represents the energy dissipation per cycle, are used in the calculations. The phase angle ${\displaystyle \phi }$, storage modulus G', and loss modulus G then become:

${\displaystyle \phi =\sin ^{-1}{\frac {4A}{\pi \Delta T_{s}\Delta d}}}$

${\displaystyle G'={\frac {\Delta T_{s}}{\Delta d}}\cos \phi }$

${\displaystyle G''={\frac {\Delta T_{s}}{\Delta d}}\sin \phi ={\frac {4A}{\pi \omega \Delta d^{2}}}}$

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.

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.[23] 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.[24]

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.[25] Strong correlations between blood viscoelasticity and regional and global cerebral blood flow during cardiopulmonary bypass have been documented.[26]

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.[27]

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.[28]

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.[29] 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.[30][31] Advancements in medical procedures and devices required a better understanding of the mechanical properties of blood.

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.[1]

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.[1]