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1.Photon Migration

Photon migration is the term used to remind us that photons, in this case near infrared (NIR) ones, propagate diffusely through most tissues.  Unlike x-rays, NIR photons are multiply scattered, a fact that significantly affects how we do optical imaging and spectroscopy in tissues.

Before turning to more formal mathematics, let us continue to discuss the life of a photon moving through a tissue.  For now, we will assume that the tissue is effectively homogeneous, meaning there are no appreciable inhomogeneities like layers or tumors in the optical field of view.

The path of one photon in a tissue
NIR photons scatter a lot, as depicted in the cartoon below.   A light source (left) injects photons into a tissue (blue box), only to be collected by a detector on the other side (right).  As the photons encounter structures in the tissue, i.e., index of refraction variations, the photon changes direction.  The intensity and angle of the scattering depends upon the size of the scatterer relative to the wavelength as well as the magnitude of the index mismatch (i.e, Mie Theory ... for an online calculator of this check out this work by Scott Prahl).  Eventually the photon may escape on the right of the tissue and be detected, in this case by a camera.
Cartoon showing the intense multiple scattering nature of light transport in tissues.

But how much scattering occurs?  How tortuous is the path?  A typical value of scattering in say breast tissues exposed to NIR photons is about 1 1/mm.  Actually, this is the quantity known as the reduced scattering (more on this later), but for now it is enough to know it represents a length scale over which the light transport is approximately isotropic (i.e., no preferred propagation direction).  The real microscopic scattering coefficient of tissues is normally about 10x higher, so we will use 10 1/mm.  That means the mean free path for NIR light in tissues is 1/10 mm = 100 microns, which on par with the thickness of a human hair (~ 50 microns) and the size of a mammillian cell (10-30 microns).   Our little NIR photon does not go very far in tissues without crashing into something.

The average path of the photon
However we typically send many photons (billions and billions) into a tissue, so we need to know how the ensemble behaves and not just a single photon.  We can catch a glimpse of this average pathlength by turning to theory.

Another way to describe the photon path length (L) is to relate it to the distance between the source and detector (rho).  The name for this ratio is the "Differential Pathlength Factor," or DPF for short.  If we use diffusion theory (as will be explained) in a semi-infinite medium geometry (i.e., we put the source and detector on the surface of a relatively large tissue), we can quantify the extra distance these photons travel due to tissue scattering:
We can see that expanded pathlength depends upon the optical properties of the tissue (absorption and scattering), as well as rho.  For a derivation of this expression you can look to this fine paper.  We also note that this expression assumes a steady-sate signal (i.e., no modulation of the source yet).

We can calculate the average photon pathlength for three cases of interest:
    A where absorption = 0.01 1/mm and reduced scattering = 1.0 1/mm
    B where absorption = 0.02 1/mm and reduced scattering = 1.0 1/mm
    C where absorption = 0.01 1/mm and reduced scattering = 1.5 1/mm.

Photon path lengths in tissues are dramatically increased by tissue scattering.

In case A we can see that the actual pathlength is easily 5x or more greater than the source detector separation (rho).  In breast cancer imaging, rho is about 30 mm, meaning that detected photons have really traveled over 300 mm in the tissue (talk about taking the 'long way home').  For case B, the pathlengths are shorter than for A.  This is because the added absorption will delete photons, especially those who travel a longer distance.  In case C, the pathlength is longer because more scattering keeps the photon transport going and going.  Thus, the overall signal is more dispersed over space (i.e., higher attenuation).

Dispersion in time
We can get another view of the paths in tissues by looking at how the photons move in time.  If we assume that the tissue is effectively homogeneous, then time and distance are the same quantity (i.e., proportional by the speed of light).  Because the process of light transport in the case of multiple scattering is mainly stochastic, not all the photon pathlengths will be identical.  We can think of this idea as dispersion in time; some will arrive at the detector early, some later.

Description of light transport in time through a tissue.

In the above figure we start with a pulse of light, where a burst of photons comes from a laser beam.  We represent this pulse by a narrow line in time with a high intensity (left).  Of the many photons contained within this pulse, some may scatter relatively few times; these photons are listed as green paths in the tissue.  They arrive at the detector very early (it is hard to see but the plot on the right has a small green start to it).  These photons, so-called "ballistic photons," are far too few to be useful in imaging thick tissues; hence the small detected intensity.

Other photons get bounced around a little bit (the purple path).  These photons arrive a bit later, but are also not many in number.  It is possible to image tissues using these "early gate" photons but the tissues have to be thin.  If the sample thickness is too great, or the scattering too intense, don't expect a lot of these photons to arrive at your detector.

The majority of the detected signal emerges from diffusive photons (the blue path), such as in the examples above.  These photons arrive in time in proportion to their pathlengths; photons taking a shorter stroll through the tissue come early, whereas those which probe a large volume of tissue will arrive later and fewer in number.  You can see that there is a peak in the signal, after which the signal decays over time.  These longer times represent photons that have sampled increasing volumes of tissues.

The key thing to note by this example is the broadening of the initial pulse.  We started with a sharp temporally focused beam and it wound up quite broad in time.  In tissues, this broadening is on the order of a few ns, depending upon the source-detector separation.  The practice of measuring the temporal broadening of photons propagating through a tissue is as Time Domain Spectroscopy.  The amount of broadening is determined by the same parameters that dictate the photon pathlength; source-detector separation, absorption and scattering.  We can measure the time response to the pulse and recover the tissue optical properties.

If you understand this idea, we can move on to an alternative method that does the exact same thing, but rather than looking at dispersion in time it looks at dispersion in frequency.  Hence, we move to the frequency domain method.