150
Biologically Inspired Robotics
8.2 Cell Mechanical Modeling
In our previous work (Tan, Sun, and Huang 2010; Tan et al. 2008), we proposed a theoretical model to interpret the deformation response of biological cells in microinjection. This model was based on membrane theory and
unitized a hyperelastic material to describe the deformation behavior of cell
membranes. In this chapter, according to the practical conditions of RBCs
stretching experiments by optical tweezers, the mechanical model is modified and extended to extract the mechanical properties of RBC membranes.
The experimental conditions of RBCs stretching meet the prerequisite and
restriction of the mechanical model developed in our previous work. First,
human RBCs in hypotonic solutions appear to be spherical or spheroidic; that
is, rotationally symmetric. Second, RBC biomembranes are usually treated
as incompressible homogeneous isotropic materials (Henon et al. 1999; Mills
et al. 2004). Third, during the deformation process of RBCs, their internal volumes are generally considered to stay constant; that is, cytoplasm is incompressible (Dao, Lim, and Suresh 2003; Mills et al. 2007).
According to the shell theory of Landau and Lifshitz (1986), the contribution of the bending rigidity can be neglected due to the small thickness of
biomembrane. Then, the deformation behavior of RBCs in optical stretching
is mainly determined by membrane theory. Quasistatic equilibrium equations are used to describe the force balance in the meridian tangential and
normal directions of the cell membrane, which are expressed by (Feng and
Yang 1973; Tan, Sun, and Huang 2010; Tan et al. 2008, 2009):
∂7
'
1 λ
'
∂7
+
1
1
λ
'
ρ
2 = (7
λ
2 − 7
∂
ρ
1 )
∂λ
(8.1)
1
2
K T
1 1 + K 2 T 2 = P
(8.2)
where T 1 and T 2 , λ 1 and λ 2 , and K 1 and K 2 are the principal tensions, stretch
ratios, and curvatures, respectively. The indices 1 and 2 refer to the corresponding component in the meridian and circumferential directions of the
deformed membrane, respectively. P is the external pressure acting on the
membrane in the normal direction. ρ and η are the coordinates after deformation as shown in Figure 8.1. The prime denotes the derivative with respect
to the angle ψ.
The principal tensions T 1 and T 2 are calculated according to the strain
energy function of the chosen membrane material. Because the constitutive
material proposed by Evans and Skalak (1980; ES material) is usually used to
represent the deformation behavior of RBC membranes, it is adopted here to
Biologically Inspired Robotics
8.2 Cell Mechanical Modeling
In our previous work (Tan, Sun, and Huang 2010; Tan et al. 2008), we proposed a theoretical model to interpret the deformation response of biological cells in microinjection. This model was based on membrane theory and
unitized a hyperelastic material to describe the deformation behavior of cell
membranes. In this chapter, according to the practical conditions of RBCs
stretching experiments by optical tweezers, the mechanical model is modified and extended to extract the mechanical properties of RBC membranes.
The experimental conditions of RBCs stretching meet the prerequisite and
restriction of the mechanical model developed in our previous work. First,
human RBCs in hypotonic solutions appear to be spherical or spheroidic; that
is, rotationally symmetric. Second, RBC biomembranes are usually treated
as incompressible homogeneous isotropic materials (Henon et al. 1999; Mills
et al. 2004). Third, during the deformation process of RBCs, their internal volumes are generally considered to stay constant; that is, cytoplasm is incompressible (Dao, Lim, and Suresh 2003; Mills et al. 2007).
According to the shell theory of Landau and Lifshitz (1986), the contribution of the bending rigidity can be neglected due to the small thickness of
biomembrane. Then, the deformation behavior of RBCs in optical stretching
is mainly determined by membrane theory. Quasistatic equilibrium equations are used to describe the force balance in the meridian tangential and
normal directions of the cell membrane, which are expressed by (Feng and
Yang 1973; Tan, Sun, and Huang 2010; Tan et al. 2008, 2009):
∂7
'
1 λ
'
∂7
+
1
1
λ
'
ρ
2 = (7
λ
2 − 7
∂
ρ
1 )
∂λ
(8.1)
1
2
K T
1 1 + K 2 T 2 = P
(8.2)
where T 1 and T 2 , λ 1 and λ 2 , and K 1 and K 2 are the principal tensions, stretch
ratios, and curvatures, respectively. The indices 1 and 2 refer to the corresponding component in the meridian and circumferential directions of the
deformed membrane, respectively. P is the external pressure acting on the
membrane in the normal direction. ρ and η are the coordinates after deformation as shown in Figure 8.1. The prime denotes the derivative with respect
to the angle ψ.
The principal tensions T 1 and T 2 are calculated according to the strain
energy function of the chosen membrane material. Because the constitutive
material proposed by Evans and Skalak (1980; ES material) is usually used to
represent the deformation behavior of RBC membranes, it is adopted here to