Biomedical Engineering Reference
In-Depth Information
(a)
60
37°C
50
40
30
24°C
20
G′
G′′
10
14°C
4°C
0
0
20
40
SHG fraction (%)
60
80
100
(b)
60
37°C
G′
G′′
50
40
30
24°C
20
10
14°C
4°C
0
0
500 1000
Mean segmented SHG (a.u.)
1500
FIgurE 11.7
(a) Average shear moduli (storage modulus,
G
′; loss modulus,
G
″) versus average SHG image area
fraction from 4 mg/mL acellular collagen gels polymerized at temperatures 4°C, 14°C, 24°C, and 37°C. (b) Average
shear moduli (storage modulus,
G
′; loss modulus,
G
″) versus mean segmented SHG image intensity from 4 mg/
mL acellular collagen gels polymerized at temperatures 4°C, 14°C, 24°C, and 37°C. (Reprinted from
Biophys J
., 92,
Raub, C. B. et al., Noninvasive assessment of collagen gel microstructure and mechanics using multiphoton micros-
copy, 2212-2222, Copyright 2007, with permission from Elsevier.)
where ρ is the density of polymer contour length per unit volume and
L
p
is the chain's persistence length.
A network mesh size,
L
m
, relates to ρ as ρ =
L
m
−
. A fiber's persistence length is proportional to its bend-
ing modulus, which for collagen scales with the fourth power of fiber diameter,
d
[94,95]. Recasting the
scaling relationship of Equation 11.6 in terms of mesh size and fiber diameter,
G ~
′
L
−
14 5
/
d
−
4 5
/
(11.7)
m
To determine if SHG images can provide structural data that scales appropriately with
G
′, the mesh size
and fiber diameter were estimated from SHG images (Figure 11.8a) in two ways: using particle analysis
and manual fiber diameter measurements (
P
PA
and
d
SHG
for fibers and pores, respectively), and using
image correlation spectroscopy, which create characteristic pore and fiber diameters,
P
ICS
and
d
ICS
, based
upon signal periodicity in the SHG image [22]. The data from SHG images were input into the scaling
relationship and a plot of log
G
′ versus log (
L
m
−14/5
d
−4/5
) was generated. The correlation using the par-
ticle analysis and hand-measured parameters is good (
R
2
= 0.93). The best-fit slope is of the order one
(
m
~ 0.84, Figure 11.8b). One conclusion to draw from this correlation is that mesh size and fiber diam-
eter, parameterized from SHG images, may explain most of the variation in storage modulus of these
acellular collagen gels, and that the gels behave as entangled networks of semiflexible fibers.