To extract fiber organization information from the acquired scattering patterns, it is crucial to interpret them using appropriate optics theory. We opted for a simple (but accurate) approach from among the numerous possibilities.
31 Using Fraunhofer diffraction theory, it can be shown that the intensity of the scattering pattern produced by a small rectangular aperture (of width
a, length
b, and angle φ defined in the plane of the aperture, which is perpendicular to the laser beam) is
32 where
I is the imaged intensity;
r, θ are polar coordinates measured from the center of the scattering pattern;
I0 is the incident light intensity; λ is the laser wavelength;
L is the distance between the aperture and the diffuser screen; sinc is the sine cardinal function; and
ā =
a/λ
L and
b̄ =
b/λ
L. We then used Babinet's principle, which states that the scattering pattern of a rectangular aperture is the same (except in its center) as that produced by a fiber (e.g., collagen I) of the same size and shape. Since thin, soft tissues can be considered to be fiber assemblies, and, under the assumption of linearity (as demonstrated in
24 for thin soft tissues), the scattering intensity for a fiber assembly
IFA can be obtained from
equation 1 as
where we performed the following change of variable χ = θ − φ for simplicity. For our purposes, the key quantity in
equation 2 is
P, the fiber distribution function, which describes the angular distribution of fibers responsible for each generated scattered light pattern. In other words,
equation 2 tells us that the total scattered-light intensity is the weighted linear sum (through
P) of the contribution of each fiber. Finally, under the assumption of high aspect-ratio (i.e., thin and long) fibers, which is satisfied for collagen I,
equation 2 can be considerably simplified, as proposed by McGee and coworkers,
32 as
Equation 3 is a simple result, as the term in front of
P does not depend on θ and can be regarded as a constant of proportionality. Using the normalization condition for
P (i.e.,
),),
equation 3 can be simplified to extract
P as
Equation 4 implies that the normalized light intensity distribution at any chosen radius
r =
R corresponds to that of the fibers, but shifted 90°.