Hybrid theory for photon transport in tissue

Hybrid Theory for photon transport in tissue uses the advantages and eliminates the deficiencies of both the Monte Carlo method and the diffusion theory for photon transport to model photons traveling through tissue both accurately and efficiently.
MCML (Monte Carlo Modeling of Light Transportation in Multi-Layered Medium).
The MCML is a numerical way to simulate photon transport in biological tissue. Each photon packet follows a random walk with persistence, where the direction of each step dependent on the direction of the previous step. By averaging multiple independent random walks, MCML estimates the ensemble-averaged quantities such as reflectance, transmittance, absorption, and fluence.
Briefly, a packet of photon is first launched into the biological tissue. The parameters of photon transport, including the step size and deflection angle due to scattering, are determined by random sampling from probability distributions. A fraction of weight, determined by the scattering and absorption coefficients is deposited at the interaction site. The photon packet continues propagating until the weight left is smaller than a certain threshold. If this packet of photon hits the boundary during the propagation, it is either reflected or transmitted, determined by a pseudorandom number. Statistically sufficient numbers of photon packets must be simulated to obtain the expected values accurately.
Advantages and Disadvantages
This Monte Carlo method is rigorous and flexible. However, because of its statistical nature, this method requires tracking a large number of photon packets, making it computationally expensive.
Diffusion Theory.
The Diffusion Theory is an approximation of the radiative transfer equation (RTE), and an analytical way to simulate photon transport. As such, it has the ability to model photon propagation through tissue quickly.
As an example, one way to attain a solution for a pencil beam that is vertically incident on a semi-infinite homogeneous scattering medium is by taking three approximation steps as follows:
Advantages and Disadvantages
Diffusion Theory is more computationally efficient than MCML. However, it is also less accurate than MCML near the source and boundaries.
Hybrid Theory.
The Hybrid Theory combines the Diffusion Theory and the Monte Carlo method in order to increase accuracy near the source and boundaries while reducing computation time. In the previous example for the Diffusion Theory, a semi-infinite scattering medium with only one boundary was assumed. If the geometry is a slab, the second boundary must be taken into account. The fluence rate at the extrapolated boundaries must be approximately 0. Using an array of image sources fulfills this boundary condition. The extrapolated boundary is located at distance formula_4. The formula_5 coordinates for the source pairs are formula_6 where formula_7 is the formula_5 coordinate for the point source and formula_9 is the slab thickness. Only 2-3 pairs are usually necessary to achieve good accuracy.
A Monte Carlo approach can be used to make up for the Diffusion Theory's inherently poor accuracy near the boundaries. As mentioned before, the Monte Carlo simulation is time consuming. When a photon packet is within a critical depth formula_10 the Monte Carlo simulation tracks all packets but within the center region the photon packet is transformed to an isotropic source and subsequently treated with Diffusion Theory. Just like in the Monte Carlo simulation, any photon packet that gets reemitted is added to the diffuse reflectance formula_11.
When a photon packet is scattered into the center zone formula_12, it is conditionally converted to an isotropic point source. The photon packet must still be in the center region after one transport mean free path formula_13 along the direction of the photon packet propagation for it to be converted to a point source, otherwise the Monte Carlo simulation continues. Before the conversion to an isotropic point source, the photon packet reduces its weight due to its interaction with the scattering medium. The resulting weight is recorded as a source function formula_14. This is the accumulated weight distribution which can be converted to relative source density function formula_15 by:
The additional diffuse reflectance formula_19 from the sources is calculated as:
Advantages over Diffusion Theory and MCML.
A trade-off between simulation speed and accuracy exists; choosing a critical depth formula_10 becomes the deciding factor for simulation speed with a deeper critical depth resulting in slower times due to packets needing to be tracked for a longer distance before the transition to diffusion theory.
Advantages
Where formula_26 is the relative refraction coefficient, formula_9 is the slab thickness, formula_28 is the absorption coefficient, and formula_29 is the user time.