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Correction of anisotropy coefficient in original Henyey Greenstein phase function for Monte Carlo simulations of light transport in tissue

Żołek N. S., Wojtkiewicz S., Liebert A.

Biocybernetics and Biomedical Engineering

2008

Vol. 28, no. 4

59-73

In this paper, two different methods for calculation of polar deflection angle are compared. The scattering angle is defined by numerical inversion of cumulative distribution of the original Henyey-Greenstein phase function. Results of the Monte Carlo simulations obtained in this manner are compared with results of simulations in which the analytical inversion of the probability density for the cosine of the deflection angle is applied. Investigations are carried out for media with optical properties similar to these typical for living tissues as well for very small source detector separations (50-500 [\mi]m), i.e. in conditions, in which the diffusion theory can not be applied. The distributions of visiting probability of photons penetrating into the semi-infinite medium are obtained for various methods of phase function calculation. It can be observed that the methods of calculation of polar deflection angle influence significantly spatial distributions of reflectance and visiting probability obtained by Monte Carlo simulations. The approximated transformation of the anisotropy coefficient used in simulations carried out with the use of the original Henyey-Greenstein function to effective anisotropy coefficient is presented; that makes possible comparisons of the results of Monte Carlo simulations obtained by using different methods.

Dynamic stability and spatial heterogeneity in the individual-based modelling of a Lotka-Volterra gas

Waniewski J., Jędruch W., Żołek N. S.

International Journal of Applied Mathematics and Computer Science

2004

Vol. 14, no 2

139-147

Computer simulation of a few thousands of particles moving (approximately) according to the energy and momentum conservation laws on a tessellation of 800 x 800 squares in discrete time steps and interacting according to the predator-prey scheme is analyzed. The population dynamics are described by the basic Lotka-Volterra interactions (multiplication of preys, predation and multiplication of predators, death of predators), but the spatial effects result in differences between the system evolution and the mathematical description by the Lotka-Volterra equations. The spatial patterns were evaluated using entropy and a cross correlation coefficient for the spatial distribution of both populations. In some simulations the system oscillated with variable amplitude but rather stable period, but the particle distribution departed from the (quasi) homogeneous state and did not return to it. The distribution entropy oscillated in the same rhythm as the population, but its value was smaller than in the initial homogeneous state. The cross correlation coefficient oscillated between positive and negative values. Its average value depended on the space scale applied for its evaluation with the negative values on the small scale (separation of preys from predators) and the positive values on the large scale (aggregation of both populations). The stability of such oscillation patterns was based on a balance of the population parameters and particle mobility. The increased mobility (particle mixing) resulted in unstable oscillations with high amplitude, sustained homogeneity of the particle distribution, and final extinction of one or both populations.