Biomedical Engineering Reference
In-Depth Information
where
L
is the length of the harmonophore assembly in the axial direction and Δ
k
is the phase mismatch,
which increases for more random assemblies [30]. As a result, the relative alignment of molecular
sources, fibrils, and fibers is reflected in the magnitude of the second-order nonlinear susceptibility
tensor χ
(2)
and hence in the experimentally observed SHG intensity. Moreover, the SHG directional-
ity and polarization properties contain additional subresolution information. As a consequence of the
imperfect phase matching in tissues, the SHG signal has a distribution of emitted forward and backward
components, whose ratio we denote as
F
SHG
/
B
SHG
. The extent of mismatch (and resulting directionality)
depends on the regularity of the fibril/fiber assembly, where structures that are ordered on the size of
λ
SHG
in the axial direction will give rise to predominantly forward SHG, whereas smaller structures
will be less forward directed. Therefore, SHG imaging is a valuable tool to investigate the mechanism
of optical clearing in collagen and striated muscle due to structural changes that occur in this process.
Below, we will examine the currently available experimental evidence for the mechanism of optical
clearing. Throughout this treatment, the interpretation of the experimental data will be enhanced with
theoretical simulations.
8.3 Monte carlo Simulations
In a tissue imaging experiment, the measured SHG signal will be composed of a convolution of the
initially emitted directionality (
F
SHG
/
B
SHG
; described above) as well as the subsequent scattering of these
photons at λ
SHG
. The scattering coefficient,
μ
s
, is a measure of density, where it is the inverse of the dis-
tance a photon will propagate before undergoing a scattering collision and changing direction. For most
tissues, these scattering lengths are typically ~20-50 μm in the visible/near-IR region of the spectrum.
The scattering anisotropy,
g
, is related to the directionality of the scattering, and varies from 0 to 1,
where higher values correspond to more ordered structural organization. Both the SHG creation and
propagation properties undergo changes in the clearing process, as fibril/fiber assembly will be effec-
tively altered by OCA, which will change the χ
(2)
tensor and consequently the scattering parameters.
The sources of forward- and backward-propagating photons (i.e., from direct emission or from multiple
scattering) cannot be determined experimentally, and a Monte Carlo simulation approach is used to
decouple these processes. Similarly, the experimentally measured dependence of SHG intensity as a
function of depth (i.e., the attenuation) into the tissue cannot be fitted with the single exponential as
it represents convolution of two depth-dependent “filter” effects. In general, the combination of three-
dimensional (3D) SHG imaging with Monte Carlo simulations of experimental data is a powerful tool
for revealing additional subresolution information, as discussed in detail in Chapters 1 and 6.
In the case of optical clearing, the simulations give us insight into the clearing mechanism. Our
Monte Carlo approach is based on the framework of Jacques and Wang [31], where we modified this to
include all the factors necessary to model a 3D SHG imaging. Monte Carlo simulations are based on
photon diffusion using the following bulk optical parameters as inputs: scattering coefficient
µ
s
, absorp-
tion coefficient
µ
a
, anisotropy of scattering
g
, and refractive index
n
. Since all these parameters are
wavelength dependent, they must be measured at both the fundamental and SHG wavelengths. Then,
simulation algorithm is used to trace the fate of every photon that can be scattered multiple times, and it
can be either transmitted or absorbed with certain probability. A flowchart of the simulation algorithm
is shown in Figure 8.2 [32].
The laser excitation is attenuated by scattering before it arrives at the focal plane (primary filter).
Then, SHG signal travels through the scattering medium until it exits the sample (secondary filter).
Obviously, the relative contributions of these two filters vary with depth. Experimental data alone does
not allow one to decouple the two processes, but Monte Carlo simulations provide that missing part
of the puzzle. To determine these contributions, first, optical sectioning is simulated at the excitation
numerical aperture by calculating the fraction of incident laser photons that arrives at the focal point
at a given depth and the resulting SHG efficiency is calculated based on the square of the remaining
intensity. The secondary filter effects are then modeled by calculating the propagation losses governed
