**2. Deformability of RBCs**

RBCs are the most deformable cell in the human body. RBC deformabiltiy is an intrinsic mechanical property determined by (1) its geometry, (2) cytoplasmic viscosity, mainly attributed to hemoglobin (Hb) solution in the cytoplasm, and (3) viscoelastic properties of RBC membrane cortex structure.

## **2.1. RBC geometry**

Mature human RBCs have a biconcave disc shape and they do not contain nucleus or subcellular structures but mainly consist of Hb solution in the cytoplasm (Fig. 1a). A typical human RBC has a thickness of 2-3 μm, diameter of 6-8 μm, cell volume of 90 fl, and surface area of approximately 136 μm2 (Kenneth,2010). Depending on species, RBC shape and size vary. In mammals, RBCs develope from nucleated progenitor cells in bone mellow but RBCs discard their nucleus as they mature, whereas RBCs of other vertebrates have nuclei*.* Throughout their life span of 100-120 days, human RBCs circulate the body delivering oxygen from the lungs to tissues. RBCs gradually lose their deformability with age and eventually rupture in spleen. The biconcave shape of normal RBCs has advantages in having deformability. Compared to a spherical shape, RBCs with biconcave shape have less volume for a given surface area, which can decrease bending energy associated with the membrane (Canham,1970).

Measurement Techniques for Red Blood Cell Deformability: Recent Advances 169

Elastic property chracterizes deformability of a material when a force is applied. Since RBC cytoplasm mainly consists of Hb solution, the elastic properties of RBC is determined by RBC membrane cortex structures. RBC membranes are only a few molecules thick, and they can be treated with a 2-D continuum model. Although the deformation of RBC membrane is highly complex, it can be simply explained by three fundamental deformation modes: area

Area expansion Shear Bend

**Figure 2.** Schematic illustrations of area expansion, shear, and bend modes of a 2-D membrane.

The elastic property of a 2-D membrane cortex is characterized by three mechanical elastic moduli: area expansion modulus *K*, shear modulus *µ*, and bending modulus *B*. The detailed

**Area expansion modulus**. The area expansion (or compression) modulus *K* reflects the elastic energy storage produced by an isotropic area dilation or compression of the

> *<sup>A</sup> T K A*

the increase in surface area, respectively (Hochmuth and Waugh,1987). The area expansion modulus of RBC membranes is mainly dominated by the elasticity of the bilayer. Interestingly, the lipid bilayer itself is highly inextensible; the stand-alone lipid bilayer area compression modulus was given in the range of 200-300 mN/m (Rawicz, Olbrich et al.,2000). However, RBC membranes exhibit significant area extensibility. There is a wide range of measured values for *K* of RBCs that fall into two groupings. (1) Values reported from micropipette-based studies are in the range of 300-500 mN/m (Evans,1973; Waugh and Evans, 1979). (2) Recently, measurements based on dynamic membrane fluctuations report *K* of RBC membranes in the range of 10-100 N/m (Gov, Zilman et al.,2003; Betz, Lenz et al., 2009; Park, Best et al.,2010; Park, Best et al.,2011; Byun, Higgins et al., in press). These two techniques sample mechanical responses of the RBC under very different loading conditions and they involve different components of the cell; micropipette-based studies mainly probes lipid-bilayer dominated behavior while membrane fluctuation measurements primarily analyze spectrin network dominated behavior. In addition, the area expansion modulus *K* of RBC membranes can be changed by temperature; the micropipette asiperation techniques measured *K* at 25 °C is 450 mN/m and the temperature dependency of *K* was found to be -6

*o*

*A* correpond to the isotopic tensile force, the original surface area, and

(1)

*2.3.1. Elastic property* 

where *Tt*, *A0*, and

expansion, shear, and bend of the membrane (Fig. 2).

explanations for three elastic moduli are described as follow:

, *<sup>t</sup>*

mN/m°C. (Waugh and Evans,1979).

membrane surface. The area expansion modulus *K* is described as

#### **2.2. Membrane cortex structure**

The unique deformability of RBCs is mainly determined by the structures of RBC membrane cortex. The membrane of human RBC is a multicomponent structure comprised of three layers: (1) an external carbohydrate-rich layer, (2) the phospholipid bilayer with 4-5 nm thickness, embedded with transmembrane proteins, and (3) a 2-D triangular mesh-like spectrin cytoskeleton network attached to the surface bilayers. The mesh size of the spectrin network is 60-80 nm. The spectrin network is anchored to the phospholipid bilayer via juntional complexes and ankyrin proteins. Junctional complexes and ankyrin proteins can diffuse in the lipid membrane.

**Figure 1.** (a) RBC morphology. (b) Spectrin network measured by high resolution negative staining electron microscopy. (b) Schematic model of the red cell membrane. Reproduced, with permission, from (Liu, Derick et al.,1987; Tse and Lux,1999).

#### **2.3. Viscoelastic properties of RBC**

In a view of classical mechanics, soft biomaterials can be characterized by viscoelastic properties - exhibiting both energy-storing elastic and energy-dissipating viscous characteristics. RBC is a typical soft biomaterial showing unique viscoelastic properties (Hochmuth and Waugh,1987).

#### *2.3.1. Elastic property*

168 Blood Cell – An Overview of Studies in Hematology

Mature human RBCs have a biconcave disc shape and they do not contain nucleus or subcellular structures but mainly consist of Hb solution in the cytoplasm (Fig. 1a). A typical human RBC has a thickness of 2-3 μm, diameter of 6-8 μm, cell volume of 90 fl, and surface area of approximately 136 μm2 (Kenneth,2010). Depending on species, RBC shape and size vary. In mammals, RBCs develope from nucleated progenitor cells in bone mellow but RBCs discard their nucleus as they mature, whereas RBCs of other vertebrates have nuclei*.* Throughout their life span of 100-120 days, human RBCs circulate the body delivering oxygen from the lungs to tissues. RBCs gradually lose their deformability with age and eventually rupture in spleen. The biconcave shape of normal RBCs has advantages in having deformability. Compared to a spherical shape, RBCs with biconcave shape have less volume for a given surface area, which can decrease bending energy associated with the

The unique deformability of RBCs is mainly determined by the structures of RBC membrane cortex. The membrane of human RBC is a multicomponent structure comprised of three layers: (1) an external carbohydrate-rich layer, (2) the phospholipid bilayer with 4-5 nm thickness, embedded with transmembrane proteins, and (3) a 2-D triangular mesh-like spectrin cytoskeleton network attached to the surface bilayers. The mesh size of the spectrin network is 60-80 nm. The spectrin network is anchored to the phospholipid bilayer via juntional complexes and ankyrin proteins. Junctional complexes and ankyrin proteins can

**(c)**

**Figure 1.** (a) RBC morphology. (b) Spectrin network measured by high resolution negative staining electron microscopy. (b) Schematic model of the red cell membrane. Reproduced, with permission, from

In a view of classical mechanics, soft biomaterials can be characterized by viscoelastic properties - exhibiting both energy-storing elastic and energy-dissipating viscous characteristics. RBC is a typical soft biomaterial showing unique viscoelastic properties

**2.1. RBC geometry** 

membrane (Canham,1970).

**2.2. Membrane cortex structure** 

diffuse in the lipid membrane.

**(a) (b)**

Volume ~ 90 fl Surface area ~ 136 µm2

~ 8 µm

(Liu, Derick et al.,1987; Tse and Lux,1999).

**2.3. Viscoelastic properties of RBC** 

(Hochmuth and Waugh,1987).

~ 2 µm

Elastic property chracterizes deformability of a material when a force is applied. Since RBC cytoplasm mainly consists of Hb solution, the elastic properties of RBC is determined by RBC membrane cortex structures. RBC membranes are only a few molecules thick, and they can be treated with a 2-D continuum model. Although the deformation of RBC membrane is highly complex, it can be simply explained by three fundamental deformation modes: area expansion, shear, and bend of the membrane (Fig. 2).

**Figure 2.** Schematic illustrations of area expansion, shear, and bend modes of a 2-D membrane.

The elastic property of a 2-D membrane cortex is characterized by three mechanical elastic moduli: area expansion modulus *K*, shear modulus *µ*, and bending modulus *B*. The detailed explanations for three elastic moduli are described as follow:

**Area expansion modulus**. The area expansion (or compression) modulus *K* reflects the elastic energy storage produced by an isotropic area dilation or compression of the membrane surface. The area expansion modulus *K* is described as

$$T\_t = K \frac{\Delta A}{A\_o} \,\prime \tag{1}$$

where *Tt*, *A0*, and *A* correpond to the isotopic tensile force, the original surface area, and the increase in surface area, respectively (Hochmuth and Waugh,1987). The area expansion modulus of RBC membranes is mainly dominated by the elasticity of the bilayer. Interestingly, the lipid bilayer itself is highly inextensible; the stand-alone lipid bilayer area compression modulus was given in the range of 200-300 mN/m (Rawicz, Olbrich et al.,2000). However, RBC membranes exhibit significant area extensibility. There is a wide range of measured values for *K* of RBCs that fall into two groupings. (1) Values reported from micropipette-based studies are in the range of 300-500 mN/m (Evans,1973; Waugh and Evans, 1979). (2) Recently, measurements based on dynamic membrane fluctuations report *K* of RBC membranes in the range of 10-100 N/m (Gov, Zilman et al.,2003; Betz, Lenz et al., 2009; Park, Best et al.,2010; Park, Best et al.,2011; Byun, Higgins et al., in press). These two techniques sample mechanical responses of the RBC under very different loading conditions and they involve different components of the cell; micropipette-based studies mainly probes lipid-bilayer dominated behavior while membrane fluctuation measurements primarily analyze spectrin network dominated behavior. In addition, the area expansion modulus *K* of RBC membranes can be changed by temperature; the micropipette asiperation techniques measured *K* at 25 °C is 450 mN/m and the temperature dependency of *K* was found to be -6 mN/m°C. (Waugh and Evans,1979).

**Shear modulus.** The shear modulus *µ* of a 2-D structure reflects the elastic energy storage associated with extension of the membrane surface with the same membrane area. The shear modulus *µ* is described as

$$T\_s = \frac{\mu}{2} \left( \mathcal{X}^2 - \mathcal{X}^{-2} \right)\_{\prime} \tag{2}$$

Measurement Techniques for Red Blood Cell Deformability: Recent Advances 171

*2D* can be qualitatively related to a 3-D bulk visosity of

*2D* ~ 10-10-10-9 Ns/m. Reported

(4)

(5)

*3D* ·*d* where *d* is the thickness of the 2D structure. For a typical

1985) or cell Hb concentration for both normal and sickle cells (Evans, Mohandas et al., 1984). The recent experiments (Betz, Lenz et al., 2009; Yoon, Hong et al., 2009) have also measured that the bending modulus of RBCs is of the order of 50 kbT. However, several other techniques have measured lower bending moduli of RBC membranes. Studies based on measurements of RBC membrane fluctuations reported membrane bending moduli in the range of 10 kbT (~ 10-22 J) (Brochard and Lennon,1975; Zilker, Engelhardt et al.,1987; Zilker,

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, 1987). The viscous properties of RBC membranes can be determined by 3-D

**Cytoplasmic viscosity**. The values for the 3-D viscosity of blood plasma and cytosolic Hb solutions are ~ 1 mPa·s and ~ 5 mPa·s, respectively (Cokelet and Meiselman,1968). 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 2-5 mPa·s (Yoon, Hong et al.,2009). Recently, the dynamic membrane fluctuation measurements retrieved the cytoplasmic visocity of the RBCs at physiological osmotic pressure as 5-6 mPa·s, and the cytosol viscosity increases monotonically from with

**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,1976). The 2-D

surface viscosities for lipid bilayers are of the order of 10-10-10-9 Ns/m (Waugh,1982; Evans and Yeung,1994). Considering viscous dissipation due to a 2-D membrane viscosity, the

ln 2 , <sup>2</sup> *s D <sup>T</sup>*

where *t* is time (Evans and Hochmuth,1976). Assuming the RBC membrane follows Kelvin-

<sup>2</sup> / . *c D t* 

where *tc* is the recovery time after large deformation of RBC membranes (Evans and Hochmuth,1976). Typically, *tc* ~ 0.06 s at 37C (Hochmuth, Buxbaum et al.,1980), and thus if

2

 

*t*

2 2

Ziegler et al.,1992; Park, Best et al.,2010; Park, Best et al.,2011).

cytoplasmic viscosity and 2-D membrane viscosity.

increasing osmolality (Park, Best et al.,2011).

Voigt model, Eq. (4) can be simply expressed as

modified version of the shear force from Eq. (2) is described as

*3D* ~ 103 mPa·s and *d* ~ 1-10 nm, and thus

viscosity of lipid membranes

*3D* as *2D* ~

phospholipid

lipid bilayer,

*2.3.2. Viscous property* 

where *Ts* is the shear force and is the the extension ratio (Evans,1973). Shear modulus of lipid bilayers is essentially zero due to its fluidity nature; shear modulus of RBC is mainly contributed from the spectrin network. The shear moduli of RBC membranes have been extensively measured by micropipette aspiration; the values for *µ* are in the range of 6-10 *µ* N/m (Evans and La Celle,1975; Chien, Sung et al.,1978; Waugh and Evans,1979; Evans, Mohandas et al.,1984). Techniques based on optial tweezers (Lenormand, Hénon et al.,2001; Dao, Lim et al.,2003), magnetic twisting cytometry (Puig-de-Morales-Marinkovic, Turner et al., 2007), and dynamic membrane fluctuation measurements (Park, Best et al.,2010; Park, Best et al.,2011) have also reported consistent values for *µ*. The shear modulus *µ* is sensitive to the environment condition of the membrane. The shear modulus decreased as temperature increased from 5 to 45C (Waugh and Evans,1979). Decreasing pH significantly increase the shear modulus of RBC membranes, but increasing pH above 7.2 does not cause a significant change (Crandall, Critz et al.,1978). More interestingly, bimodal distributions in the values for *µ* were observed in independently reported data (Lenormand, Hénon et al., 2001; Park, Best et al.,2010), suggesting the nonlinear stiffening of spectrin network (Park, Best et al.,2011). Malaria invasion cause significant increases in shear moduli values (Mills, Diez-Silva et al.,2007).

**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

$$M = B\left(\mathbb{C}\_1 + \mathbb{C}\_2 - \mathbb{C}\_3\right) \tag{3}$$

where M is the bending momemt. C1 and C2 are two principle curvatures, and C3 is the curvature in the stress-free state (Helfrich,1973; Evans,1974). Bending of a 2-D structure involves both area compression and expansion. For a lipid bilayer structure, the bending modulus, area expansion (or compression) modulus, and the thickness of the bilayer are related by B=h2K/4, where h is the bilayer separation distance, and K is the compressibility of the bilayer (Helfrich,1973; Evans,1974). 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,2002). The elastic bending moduli B of RBC membranes have been measured with various techniques. The values for the bending modulus measured by micropipette-based studies are in the range of 50 kbT (~ 10-19 Nm) where kb is Boltzmann constant, and T is the temperature (Evans,1983). The bending modulus B of RBCs does not significantly change with temperature (Nash and Meiselman, 1985) or cell Hb concentration for both normal and sickle cells (Evans, Mohandas et al., 1984). The recent experiments (Betz, Lenz et al., 2009; Yoon, Hong et al., 2009) have also measured that the bending modulus of RBCs is of the order of 50 kbT. However, several other techniques have measured lower bending moduli of RBC membranes. Studies based on measurements of RBC membrane fluctuations reported membrane bending moduli in the range of 10 kbT (~ 10-22 J) (Brochard and Lennon,1975; Zilker, Engelhardt et al.,1987; Zilker, Ziegler et al.,1992; Park, Best et al.,2010; Park, Best et al.,2011).

#### *2.3.2. Viscous property*

170 Blood Cell – An Overview of Studies in Hematology

modulus *µ* is described as

Diez-Silva et al.,2007).

where *Ts* is the shear force and

**Shear modulus.** The shear modulus *µ* of a 2-D structure reflects the elastic energy storage associated with extension of the membrane surface with the same membrane area. The shear

> 2 2 , <sup>2</sup> *<sup>s</sup> <sup>T</sup>*

lipid bilayers is essentially zero due to its fluidity nature; shear modulus of RBC is mainly contributed from the spectrin network. The shear moduli of RBC membranes have been extensively measured by micropipette aspiration; the values for *µ* are in the range of 6-10 *µ* N/m (Evans and La Celle,1975; Chien, Sung et al.,1978; Waugh and Evans,1979; Evans, Mohandas et al.,1984). Techniques based on optial tweezers (Lenormand, Hénon et al.,2001; Dao, Lim et al.,2003), magnetic twisting cytometry (Puig-de-Morales-Marinkovic, Turner et al., 2007), and dynamic membrane fluctuation measurements (Park, Best et al.,2010; Park, Best et al.,2011) have also reported consistent values for *µ*. The shear modulus *µ* is sensitive to the environment condition of the membrane. The shear modulus decreased as temperature increased from 5 to 45C (Waugh and Evans,1979). Decreasing pH significantly increase the shear modulus of RBC membranes, but increasing pH above 7.2 does not cause a significant change (Crandall, Critz et al.,1978). More interestingly, bimodal distributions in the values for *µ* were observed in independently reported data (Lenormand, Hénon et al., 2001; Park, Best et al.,2010), suggesting the nonlinear stiffening of spectrin network (Park, Best et al.,2011). Malaria invasion cause significant increases in shear moduli values (Mills,

**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

where M is the bending momemt. C1 and C2 are two principle curvatures, and C3 is the curvature in the stress-free state (Helfrich,1973; Evans,1974). Bending of a 2-D structure involves both area compression and expansion. For a lipid bilayer structure, the bending modulus, area expansion (or compression) modulus, and the thickness of the bilayer are related by B=h2K/4, where h is the bilayer separation distance, and K is the compressibility of the bilayer (Helfrich,1973; Evans,1974). 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,2002). The elastic bending moduli B of RBC membranes have been measured with various techniques. The values for the bending modulus measured by micropipette-based studies are in the range of 50 kbT (~ 10-19 Nm) where kb is Boltzmann constant, and T is the temperature (Evans,1983). The bending modulus B of RBCs does not significantly change with temperature (Nash and Meiselman,

curvature. The bending modulus *B* of a 2-D membrane is described as

(2)

is the the extension ratio (Evans,1973). Shear modulus of

*M* <sup>123</sup> *BC C C* (3)

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, 1987). The viscous properties of RBC membranes can be determined by 3-D cytoplasmic viscosity and 2-D membrane viscosity.

**Cytoplasmic viscosity**. The values for the 3-D viscosity of blood plasma and cytosolic Hb solutions are ~ 1 mPa·s and ~ 5 mPa·s, respectively (Cokelet and Meiselman,1968). 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 2-5 mPa·s (Yoon, Hong et al.,2009). Recently, the dynamic membrane fluctuation measurements retrieved the cytoplasmic visocity of the RBCs at physiological osmotic pressure as 5-6 mPa·s, and the cytosol viscosity increases monotonically from with increasing osmolality (Park, Best et al.,2011).

**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,1976). The 2-D viscosity of lipid membranes *2D* can be qualitatively related to a 3-D bulk visosity of phospholipid *3D* as *2D* ~*3D* ·*d* where *d* is the thickness of the 2D structure. For a typical lipid bilayer, *3D* ~ 103 mPa·s and *d* ~ 1-10 nm, and thus *2D* ~ 10-10-10-9 Ns/m. Reported surface viscosities for lipid bilayers are of the order of 10-10-10-9 Ns/m (Waugh,1982; Evans and Yeung,1994). Considering viscous dissipation due to a 2-D membrane viscosity, the modified version of the shear force from Eq. (2) is described as

$$T\_s = \frac{\mu}{2} \left(\lambda^2 - \lambda^{-2}\right) + 2\eta\_{2D} \frac{\partial \ln \mathcal{X}}{\partial t} \,, \tag{4}$$

where *t* is time (Evans and Hochmuth,1976). Assuming the RBC membrane follows Kelvin-Voigt model, Eq. (4) can be simply expressed as

$$
\hbar\_c = \eta\_{2D} \mid \mu. \tag{5}
$$

where *tc* is the recovery time after large deformation of RBC membranes (Evans and Hochmuth,1976). Typically, *tc* ~ 0.06 s at 37C (Hochmuth, Buxbaum et al.,1980), and thus if *µ* ~ 1-10 *µ*N/m,  *~* 0.06 – 0.6 *µ*Ns/m. The 2-D surface viscosity of RBC membranes has been measured by several experiments. Tether experiments performed on model membrane systems, where cytoskeleton structure was absent, obtained a resultant upper bound of 5×10- <sup>3</sup> *µ*N·s/m for *2D* (Waugh,1982). The diffusion constant of membrane-bound proteins can be used to calculate the membrane viscosity (Saffman and Delbrück,1975). Using this method, the 2-D membrane viscosity values of RBC membranes have been reported in the range of (0.5-14)x10-9 Ns/m with various technqiues including fluorescence photobleaching recovery (Golan and Veatch,1980), fluorescence photo-bleaching technique (Kapitza and Sackmann, 1980), and restriction of the lateral motion of membrane embedded proteins (Tsuji and Ohnishi,1986).

Measurement Techniques for Red Blood Cell Deformability: Recent Advances 173

**Viscosity (mPas)**

**Hematocrit (%)**

**0 10 20 30 40 50 60**

**0.11 s-1 0.51 s-1 2.0 s-1 20 s-1**

Whole blood is a two-phase liquid consisting of a liquid medium (plasma) and formed elements such as RBCs, white blood cells, and platelets. Thus, its viscosity is mainly determined by (1) viscous properties of plasma, (2) the fraction of RBCs in the blood (hematocrit, normal range is 42 – 47%), and (3) viscoelastic properties of the formed

> **Deformation Elongation and orientation**

**VC-Veins Ven. AO-Art. Arterioles Cap.**

**Figure 3.** (a) Apparent viscosity of blood as a function of shear rates. (b) Hematocrit effects on blood as a function of shear rates. Modified, with permission, from (Somer and Meiselman,1993; Baskurt,2007)

 Plasma is a Newtonian fluid which viscosity in normal condition varies 1.10 ~ 1.35 cP at 37°C, while the viscosity of pure water is 1.0 cP at 20°C (Lowe, Drummond et al.,1980). Plasma proteins such as fibrinogen are thought to cause RBC aggregation by facilitating binding between RBCs. Elevated levels of fibrinogen concentration in plasma enhance

 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,1970; Baskurt,2007).

Microcirculation transports blood to the small vessels in the vasculature embedded within organs. The arterial side of vessels in the microcirculation, surrounded by smooth muscle cells, has the inner diameter of ~ 10 – 100 μm. Capillaries, parts of the microcirculation, have only one RBC thick, having the diameter of ~ 5 – 10 μm. Blood flow in microcirculation has low Reynold number and thus it is governed by Stoke's law (Baskurt,2007). 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,

**(b) (a)**

**0.1 1 10 100 1000**

**Shear rate (s-1)**

RBC aggregation and thus it increases blood viscosity.

elements.

**Aggregation**

**Normal plasma Normal red cells in plasma Hardened red cells in plasma Normal red cells in fibrinogen-free plasma**

**3.2. Blood flow in microcirculation** 

**1**

**10**

**Viscosity (mPas)**

**100**
