posted July 15, 2015. S. L. Jacques

A simple convenient approximate analytic expression describes the 1D fluence rate distribution, φ(z) [W/cm^{2}], within a thick homogeneous medium (or tissue) when irradiated with a broad uniform irradiance. The total diffuse reflectance, R_{d}, is also specified by a simple expression. |

For a homogeneous medium (or tissue) that is sufficiently thick that its rear boundary has no effect on the light distribution (i.e., equivalent to semi-infinite medium), the fluence rate (φ [W/cm^{2}]) within the medium in response to an irradiance (E [W/cm^{2}]) as a function of depth (z [cm]) can be approximated by a simple expression:

Equation 1:

φ / E = k exp(-z/δ) - (k - φ(0)/E) exp(-z/δ_{2})

and

R_{d} = 0.96exp(-7.61 / sqrt(3(1 + (µ_{s}'/µ_{a}))))

where

k = 1 + 2.52(1 - exp(-9.01 R_{d})) + 4.75 R_{d}

φ(0)/E = 1 + 6.89 R_{d}

δ_{2} = 0.657 / (µ_{s}' + 5.76 µ_{a})

µ_{a} = absorption coefficient [cm^{-1}]

µ_{s}' = reduced scattering coefficient [cm^{-1}] = µ_{s}(1-g)

µ_{s} = scattering coefficient [cm^{-1}]

g = anisotropy of scattering [dimensionless]

This Equation 1 is based on Monte Carlo simulations for a range of optical properties, for the case of a medium refractive index n = 1.37 (like a 70% water content tissue), bounded by air (n = 1.0). The Monte Carlo simulations are discussed in the next section. This report updates an earlier report,

Jacques SL, 1992, Simple optical theory for light dosimetry during PDT,

SPIE Proc. 1645, 155-165.

by adding the term (k - φ(0)/E) exp(-z/δ_{2}) to the expression for φ, which accounts for the loss of photons near the air/medium surface boundary. Figure 1 illustrates a typical example.

Monte Carlo simulations (using MCML code) generated values of fluence rate versus depth, φ(z), and diffuse reflectance, R_{d}, for a range of absorption and reduced scattering values (µ_{a} = 0.01 to 10 cm^{-1}, µ_{s}' = 5 to 100 cm^{-1}). The simulation results were then fit by analytic expressions to provide the approximate but convenient Eq. 1 for predicting φ(z), as well as the diffuse reflectance R_{d}.

The predicted diffuse reflectance, R_{d} is a function of the ratio µ_{s}' / µ_{a}, as in Eq. 1. Comparison of the predicted R_{d} versus the Monte Carlo R_{d.mc} is shown in Fig. 2:

Monte Carlo simulations generated a set of φ(z) curves like Figure 1. Fitting the curves for the factor k yielded Figure 3:

Fig. 3: Backscatter parameter k versus the diffuse reflectance, R_{d}. If there is no reflectance, k = 1.If there is reflectance, the fluence accumulates near the surface and k increases. |

The fluence rate at the tissue surface (just within the surface boundary) is dependent on the value of R_{d}. If R_{d} = 0, φ(0) / E = 1. If there is diffuse reflectance, the φ(0) / E exceeds 1.

Fig. 4: The fluence rate at the surface, φ(0), normalized by the irradiance, E, is dependent on the value of R_{d}. |

The correction for photon loss near the medium's surface is dependent on the parameter δ_{2} [cm], which depends on the optical properties as shown in Fig. 5:

Figure 6 shows all the Monte Carlo simulations of φ(z) / E.