Biomedical Engineering Reference
In-Depth Information
regardless of whether the excitation energy is significantly off-resonance. In this case, Equation A.6
becomes
2
cos
2
ϕ
tan
2
θ
=
(A.8)
I
I
y
x
sin(
2
ϕ
)
−
sin ( )
2
ϕ
And a measurement of directional SHG intensities
I
y
and
I
x
at a known φ allows the deduction of θ,
the polar angle of the SHG scatterers within a given fibril, that is, the helical pitch angle for collagen (B,
right image
). These directional SHG intensities relate to the total SHG intensity measured with a perpen-
dicularly polarized illumination beam (
I
p
) as [27]
2
I
( )
ϕ
=
I
ρ
cos
2
ϕ
+
sin
2
ϕ
y
p
2
I
( )
ϕ
=
I
sin
2
ϕ
x
p
where the fitting parameter ρ is used as a measure of the axial polarizing effects of the fibril. Thus
tan
2
θ = 2/ρ
Hence, we see that the ρ term of Williams et al. and others [27,38,69] can also give us θ, the polar
angle of the SHG scatterers in a given fibril.
In further support, SHG-based estimations of this θ [36,37,39] have correlated well with x-ray diffrac-
tion measurements of the pitch angle for a single collagen helix [70,71]. Furthermore, a very recent elegant
study suggests that when making SHG-based measurements of θ in collagen, accounting for the distinct
pitch angles of both peptide and methylene groups in single-collagen helices will yield θ values that best
agree with gold standard x-ray diffraction data, versus approaches that assume that SHG-measured θ
values arise principally from peptide contributions alone [65].
We also point out that using slightly different mathematical modeling, the polar angle of the SHG
scatterers in a given fibril, rather than providing the helical pitch angle as above, can inform us as to the
“tilt” angle of individual collagen triple helices away from the fibril axis (θ
t
−
h
, in Figure A.3) (for details,
see Ref. [66]).
Fibril
θ
t
-
h
ψ
z
y
x
FIgurE A.3
Model 2, θ as triple-helix tilt angle. (Reprinted with permission from Tuer, A.E. et al. 2011.
Nonlinear optical properties of type I collagen fibers studied by polarization dependent second harmonic genera-
tion microscopy.
J Phys Chem B
. Copyright 2011, American Chemical Society.)
