Biomedical Engineering Reference
In-Depth Information
gels, showing that SHG from in-plane fiber cross-sections is more intense that from fibers orientated
with long axes perpendicular to the image plane.
The theoretical orientation dependence of SHG signal was determined by utilizing the previously
described expression for SHG intensity,
I
2ω
[86,89]
2
z
+
∫
L
0
p
n n
e
i kz
∆
(11.3)
I
=
(
I
)
2
d
2
dz
,
2
ω
ω
eff
2
1
+
iz z
/
R
2
ω ω
z
0
where in this case,
p
is a lumped term of fundamental constants and beam parameters,
n
2
ω
and
n
ω
are
the index of refraction at the SHG and fundamental wavelengths,
I
ω
is the laser intensity at the focal
point,
d
ef
is collagen's orientation-dependent effective second-order nonlinear susceptibility,
z
R
is the
Rayleigh distance, and Δ
k
is the phase mismatch. Assuming that
p
,
n
ω
,
n
2
ω
,
z
R
, and Δ
k
remain constant
over any fiber orientation, the ratio of SHG intensities from fibers nearly perpendicular (
I
2
⊥
) versus
fibers nearly parallel (
I
2
=
) to the laser propagation direction is then
I
⊥
/
I
=
=
(
d
⊥
)
2
/
(
d
=
) ,
2
(11.4)
2
ω
2
ω
eff
eff
where
d
ef
⊥
and
d
ef
=
are the effective second-order nonlinear susceptibilities of collagen fibers perpen-
dicular and parallel to the laser propagation direction, respectively. For parallel polarized laser light, the
nonlinear susceptibility
d
ef
is [90,91]
d
=
3
16
d
(cos
β
cos
δ
−
cos
3
β
cos
3
δ
)
+
d
cos
3
β
cos
3
δ
,
(11.5)
eff
22
where β is the angle of the (assumed randomly oriented) fiber axis with respect to the fundamental
electric field and δ is the angle between the fiber axis and the imaging plane. The case of circularly
polarized laser light corresponds to allowing β to vary between 0° and 360°, and averaging
d
ef
2
over all
values of β for a specific value of δ between 0° (fiber perpendicular to laser propagation) and 90° (fiber
parallel to laser propagation). The ratio
d
22
/
d
16
, necessary to calculate
d
ef
was estimated to be ~2, based
upon previous studies in collagen [89]. In this study, experimentally determined ratios of
I
⊥
/ were
compared to these calculations to determine their reasonableness. This ratio was calculated by carrying
out a fiber segmentation, using TPF signal, into circular (out-of-plane) fiber cross-sections
c
, and ellipti-
cal (in-plane) fiber cross-sections
e
, and then determining the ratio of colocalized SHG signal in these
segmented fiber regions.
Fibers parallel and perpendicular to the image plane are visible in TPF images (see Figures 11.1a,
and 11.1i) and can be segmented after thresholding based upon particle circularity (Figure 11.5a, in-
plane fibers based upon TPF signal from coarse-structured gel similar to that of Figure 11.1a; 11.5b,
out-of-plane fibers based upon TPF signal). In contrast, only fibers with scattering interfaces more or
less parallel to the image plane (and perpendicular to the laser propagation direction) produce strong
backward-detected SHG signal (see Figures 11.1a, 11.1i). TPF
e
/TPF
c
± SE was 1.1 ± 0.1 for the pH 5.5
condition and 0.90 ± 0.05 for the pH 6.5 condition; SHG
e
/SHG
c
± SE was 3.1 ± 1.3 for the pH 5.5 condi-
tion and 2.9 ± 0.8 for the pH 6.5 condition (Figure 11.5c).
The squared effective nonlinear susceptibility of collagen was calculated using Equations 11.3 through
11.5 and was plotted as a function of δ, the angle of the fiber axis with respect to the image plane (Figure
11.5d). A ratio of 3 for
SHG
e
/
SHG
c
, for example, corresponds to
d
ef
2
values of 2.64 and 0.88 for fibers
oriented at δ = 10° and 63°, respectively (Figure 11.5d). These example values were chosen to show that
the theoretical estimation of the ratio of
d
ef
2
based upon fibers tilted at shallow versus steep angles tends
to predict an SHG intensity ratio similar to experimentally determined values.
I
2
ω
2
ω
