@article{carp04,
  author = {Stefan A. Carp and Scott A. Prahl and Vasan Venugopalan},
  title = {Radiative Transport in the $\delta$-{P}$_1$ Approximation: Accuracy of Fluence Rate and Optical Penetration Depth Predictions in Turbid Semi-Infinite Media},
  journal = {Journal of Biomedical Optics},
  pages = {632--647},
  volume = {9},
  year = {2004},
  abstract = {Using the $\delta$-P1 approximation to the Boltzmann transport equation we have developed analytic solutions for the fluence rate produced by planar (1-D) and Gaussian beam (2-D) irradiation of a homogeneous, turbid, semi-infinite medium. To assess the performance of these solutions we compare the predictions for the fluence rate and two metrics of the optical penetration depth with Monte Carlo simulations. We provide results under both refractive-index matched and mismatched conditions for optical properties where the ratio of reduced scattering to absorption lies in the range $0\le (\mu_s'/\mu_a) \le 10^4$. \\[3mm] For planar irradiation, the $\delta$-P$_1$ approximation provides fluence rate profiles accurate to $\pm$16\% over the full range of optical properties for depths up to 6 transport mean free paths (l$^*$). Metrics for optical penetration depth are predicted with an accuracy of $\pm$4\%. For Gaussian irradiation using beam radii $r_0 \ge 3l^*$, the accuracy of the fluence rate predictions is no worse than the planar irradiation case. For smaller beam radii, the predictions degrade significantly. Specifically for media with $(\mu_s'/\mu_a)=1$ irradiated with a beam radius of $r_0 = l^*$, the error in the fluence rate approaches 100\%. Nevertheless, the accuracy of the optical penetration depth predictions remains excellent for Gaussian beam irradiation, and degrades to only $\pm$20\% for $r_0 = l^*$. These results show that for a given set of optical properties $(\mu_s'/\mu_a)$, the optical penetration depth decreases with a reduction in the beam diameter. Graphs are provided to indicate the optical and geometrical conditions under which one must replace the $\delta$-P$_1$ results for planar irradiation with those for Gaussian beam irradiation to maintain accurate dosimetry predictions.},
}