## Simple light penetration into and reflectance from a slab of light-scattering medium

posted July 15, 2015. S. L. Jacques

 A simple convenient approximate analytic expression describes the 1D fluence rate distribution, φ(z) [W/cm2], within a thick homogeneous medium (or tissue) when irradiated with a broad uniform irradiance. The total diffuse reflectance, Rd, 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/cm2]) within the medium in response to an irradiance (E [W/cm2]) 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/δ2)

and

Rd = 0.96exp(-7.61 / sqrt(3(1 + (µs'/µa))))

where
k = 1 + 2.52(1 - exp(-9.01 Rd)) + 4.75 Rd
φ(0)/E = 1 + 6.89 Rd
δ2 = 0.657 / (µs' + 5.76 µa)
µa = absorption coefficient [cm-1]
µs' = reduced scattering coefficient [cm-1] = µs(1-g)
µs = 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/δ2) to the expression for φ, which accounts for the loss of photons near the air/medium surface boundary. Figure 1 illustrates a typical example.

 Fig. 1: Example of 1D light penetration into a homogeneous slab of semi-infinite thickness. The fluence rate per irradiance, φ/E [dimensionless], versus depth, z [cm], is plotted in red. The exponential k exp(-z/δ) is plotted as dashed blue line. The second term, (k - φ(0)/E) exp(-z/δ2), accounts for the loss of light at the air/medium surface boundary.The optical properties of this example are typical for red light penetration into soft tissues.

## Monte Carlo simulations that guided Eq. 1

Monte Carlo simulations (using MCML code) generated values of fluence rate versus depth, φ(z), and diffuse reflectance, Rd, for a range of absorption and reduced scattering values (µa = 0.01 to 10 cm-1, µs' = 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 Rd.

The predicted diffuse reflectance, Rd is a function of the ratio µs' / µa, as in Eq. 1. Comparison of the predicted Rd versus the Monte Carlo Rd.mc is shown in Fig. 2:

 Fig. 2(LEFT): Rd versus ratio µs' / µa. Blue circles are the Monte Carlo simulation data. Red line is equation 1. Fig. 2(RIGHT): (TOP) Comparision of Rd in Eq. 1 and Rd.mc from Monte Carlo simulations.(BOTTOM) Residual errors.

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, Rd. 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 Rd. If Rd = 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 Rd.

The correction for photon loss near the medium's surface is dependent on the parameter δ2 [cm], which depends on the optical properties as shown in Fig. 5:

 Fig. 5: The parameter δ2 specifies the 1/e depth to which loss of photons at the surface decreases the fluence rate near the surface, φ(z near surface). The median value of δ2 for the range of optical properties used in the simulations was 0.0304 cm.

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

 Fig. 6: (TOP) The fits for all the Monte Carlo simulations.The (red symbols) indicate Monte Carlo φ(z) / E. The blue lines indicate the fits using Eq. 1.(BOTTOM) The residual errors are usually less than 10%.