Biomedical Engineering Reference
In-Depth Information
2
2
2
F
a
c
+
a
(10-9)
w t
=
log
+
2
π
pR
2
2
c
a
which is linear (ϕ ∝ ω 0 ). It is interesting to compare the model with the
existing theories. Yao et al. 17 assumes a pure stretching model, ignores
the
σ 0 contribution and derives Eq. 10-2 . Comparing with the present
model, the following are noted: (i) R and c are used in Eq. 10-2 and both
are linear, but a and c are squared i n Eq. 10-9 ; (ii)
σ m is ignored and thus
a non-cubic region of F ( w t ). It is also worthy of pointing out that the
following terms are ignored by Boulbitch et al . 18 in the Eqs. 10-4 and
10-5 : (i) I 0 (βξ) in ω(ξ); (ii) log(ζ) in ω 0 ; (iii) σ m ; (iv) the dependence of a
upon F .
3.1.4. Viscoelastic model
In general, most cells under external load behave like a viscoelastic body
rather than a purely elastic entity. Hence it is sensible to construct a
theoretical model of cell indention based on viscoelasticity. Several
research groups have developed such viscoelastic models for describing
cells under indentation.
Darling et al. 22 have a simple model which incorporates viscoelasticity
of a standard linear solid into the Hertz equation to account for small
deformation of an isotropic, incompressible solid sphere indented by a
hard, spherical indenter. The model has been successfully applied for
characterizing the viscoelastic properties of zonal articular chondrocytes.
The concise form of the final constitutive relation of the cell is given by
1/ 2
2 / 3
4
Rh E
t
Ft
()
R
1
exp
(10-10)
3(1
)
where F ( t ) is the time-dependent indentation force ( i.e. , stress-
relaxation), h is the indentation depth, E R is the relaxation elastic
modulus, τ σ and τ ε are the relaxation times under constant load and
constant strain, respectively. 23
Roca-Cusachs et al . 24 have developed an alternative viscoelastic
model for AFM-indentation of neutrophils. Based on a cell cortex
 
 
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