Henyey-Greenstein Approximation

The Henyey-Greenstein phase function [73] is often used to characterize the angular distribution of scattered light by tissue and is characterized by the average cosine of the scattering angle, g. It is useful to compare the FDTD phase functions to the Henyey-Greenstein phase function to determine the accuracy of the Henyey-Greenstein approximation for cells, since it is considerably simpler.

Since the Henyey-Greenstein phase function is a probability density function, it is normalized to an area of one. To compare the computed scattering pattern to the Henyey-Greenstein phase function, $p_{hg}(\theta )$, the FDTD scattering pattern was normalized to obtain a phase function with Equation 4.61, and the anisotropy g, was computed from $p(\theta)$ and used in the Henyey-Greenstein formula,

$$p_{hg}=\frac{1}{2}\frac{1-g^2}{\left(1-2g\cos\theta+g^2\right)^{3/2}}$$ (4.72)

  

Figure 4.24: Comparison of the FDTD computed phase function with the Henyey-Greenstein approximation for two cells with differing anisotropy values.

A comparison of the FDTD and Henyey-Greenstein phase functions for two inhomogeneous cells is plotted in Figure 4.24. For one of the cells, g=0.9977 and g=0.9866 for the other cell. Figure 4.24 demonstrates that the anisotropy may not be sufficient to fully characterize the phase function since the slight decrease in g does not result in a substantial change in the Henyey-Greenstein phase functions at high angles, while there is almost an order of magnitude difference in the FDTD phase functions of the two cells.

In order to analyze the agreement of the FDTD phase functions with the Henyey-Greenstein phase function, it is useful to examine some of the higher order moments of the phase functions. Any phase function can be written in terms of a series of Legendre polynomials [20]

$$p(\cos\theta)=\frac{1}{4\pi}\sum\limits_{n=0}^{\infty}(2n+1)b_n P_n(\cos\theta)$$ (4.73)

where $P_n(\cos\theta)$ is the Legendre polynomial of order n. The phase function has been expressed in terms of $p(\cos\theta)$for ease of notation and is subject to the same normalization condition,

$$\int\limits_{-1}^{1}p(\cos\theta)d(\cos\theta)=1.$$ (4.74)

Since the Legendre polynomials are a complete set of orthogonal functions, the expansion coefficients, b_n, can be obtained from the inner product of the phase function, $p(\cos\theta)$, and the corresponding Legendre polynomial [80],

 

$$\left< p(\cos\theta), P_n(\cos\theta) \right>$$ b_n = (4.75)

$$\int\limits_{-1}^{1}p(\cos\theta)P_n(\cos\theta)d(\cos\theta).$$ = (4.76)

A variety of phase functions can be obtained by proper selection of the expansion coefficient. The Henyey-Greenstein is one such example, and for this particular case,

 

b_n=gⁿ (4.77)

Regardless of the form of the expansion coefficients, the first two terms are determined by the normalization condition of Equation 4.74 and are given by [81],

 

b₀ = 1 (4.78)

b₁ = g. (4.79)

To assess the accuracy of the Henyey-Greenstein for cell scattering, the expansion coefficients, b_n, were computed using Equation 4.75 for the FDTD phase functions plotted in Figure 4.24 and are compared to the Henyey-Greenstein coefficients (b_n=gⁿ) for n=1-4 in Table 4.2. If the Henyey-Greenstein formula was the true phase function of the cells, the following relationship should hold for the computed values based on Equation 4.77,

$$\left(b_n\right)^{1/n} = g.$$ (4.80)

 

Table 4.2: Expansion coefficients of the FDTD phase functions for comparison with the Henyey-Greenstein coefficients.

		gn	bn	b1/n
cell with $9\: \mu\text{m}$,	n=1	0.9866	0.9866	0.9866
inhomogeneous nucleus	n=2	0.9733	0.9655	0.9826
	n=3	0.9602	0.9386	0.9791
	n=4	0.9473	0.9082	0.9762
cell with $2.5\: \mu\text{m}$,	n=1	0.9977	0.9977	0.9977
inhomogeneous nucleus	n=2	0.9954	0.9938	0.9969
	n=3	0.9931	0.9884	0.9961
	n=4	0.9908	0.9820	0.9955

 

From Table 4.2 it is evident that this relationship is not entirely accurate and that the value of (b_n)^1/n converges to a value slightly less than g. If this slightly lower value of g is used in the Henyey-Greenstein formula, the phase function becomes slightly more isotropic, but it does not significantly improve the agreement with the FDTD phase functions.

An alternative to the Henyey-Greenstein, proposed by Irvine [82], uses a weighted combination of two Henyey-Greenstein phase functions with different anisotropy values,

$$p(\theta)=f p_{hg}(g_1) + (1-f)p_{hg}(g_2)$$ (4.81)

where f is a weighting factor. Note that as $g_2\rightarrow0$, $p(\theta)$ becomes the modified Henyey-Greenstein phase function [46].

Equation 4.81 was used to approximate cell scattering and is compared with the single term Henyey-Greenstein and FDTD phase functions in Figure 4.25. The cell ( $14\: \mu\text{m}$diameter) contained organelles at a volume fraction of 0.05 and a $6\: \mu\text{m}$ diameter, inhomogeneous nucleus.

  

Figure 4.25: Comparison of FDTD, single term Henyey-Greenstein, and two term Henyey-Greenstein phase functions.

The parameters used in the two term Henyey-Greenstein are: g₁=0.9844, g₂=0.05, f=0.99. For the cell used in Figure 4.25, the two term Henyey-Greenstein provides a closer match to the FDTD phase function that the single Henyey-Greenstein. However, for many cells, particularly cells that are relatively homogeneous with g>0.99, the two term Henyey-Greenstein does not provide significant improvement over the traditional Henyey-Greenstein. Therefore, for many cells, one or two parameters are insufficient to describe the phase function at all angles.

The Henyey-Greenstein phase function has been found to describe the angular pattern of scattering from tissue by several authors [46,83]. In these measurements, thin slices of tissue were used with a goniometer. In all of the measurements, however, the thickness of the tissue slices exceeded typical cell diameters ( $20\: \mu\text{m}$), and therefore cannot be directly compared to scattering from single cells. The scattering patterns of the tissue slices are the result of scattering from more than one cell layer, as well as other tissue structures such as collagen and connective tissue. Although the measured patterns of these authors were highly forward peaked, they involved a series of multiple scattering that was primarily forward directed.