Hybrid theory for photon transport in tissue
Encyclopedia
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.
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.
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.
. The 
coordinates for the source pairs are 
where 
is the 
coordinate for the point source and 
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 
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 
.
When a photon packet is scattered into the center zone
, 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 
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 
. This is the accumulated weight distribution which can be converted to relative source density function 
by:
where 
is the grid volume and 
is the number of photon packets.
The additional diffuse reflectance
from the sources is calculated as:
where 
is from the diffusion theory approximation for a slab and 
is the azimuthal angle. The total diffuse reflectance would be 
and 
added together.
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 
is the relative refraction coefficient, 
is the slab thickness, 
is the absorption coefficient, and 
is the user time.
MCML (Monte Carlo Modeling of Light Transportation in Multi-Layered Medium)
The MCMLMonte Carlo method for photon transport
Modeling photon propagation with Monte Carlo methods is a flexible yet rigorous approach to simulate photon transport. In the method, local rules of photon transport are expressed as probability distributions which describe the step size of photon movement between sites of photon-tissue interaction...
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
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
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 TheoryRadiative transfer equation and diffusion theory for photon transport in biological tissue
Photon transport in biological tissue can be equivalently modeled numerically with Monte Carlo simulations or analytically by the radiative transfer equation . However, the RTE is difficult to solve without introducing approximations. A common approximation summarized here is the diffusion...
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:
- The anisotropically scattering medium is converted to an isotropically scattering medium. That is, the scattering coefficient is scaled by
, where 
is the anisotropy. The anisotropy 
is then set to zero; - The unit-power pencil beam is converted into an equivalent isotropic point source at a depth that is equal to the transport mean free path, with a power equal to the transport albedo;
- The boundary effect of the scattering medium is removed by adding an image source to satisfy the boundary condition.
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. The coordinates for the source pairs are where is the coordinate for the point source and is the slab thickness. Only 2-3 pairs are usually necessary to achieve good accuracy. 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 .When a photon packet is scattered into the center zone
, 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 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 . This is the accumulated weight distribution which can be converted to relative source density function by: where is the grid volume and is the number of photon packets.The additional diffuse reflectance
from the sources is calculated as: where is from the diffusion theory approximation for a slab and is the azimuthal angle. The total diffuse reflectance would be and added together.Advantages over Diffusion Theory and MCML
A trade-off between simulation speed and accuracy exists; choosing a critical depth 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
- More accurate than the Diffusion Theory, especially near the source
- Faster than the Monte Carlo method
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|---|---|---|---|---|---|
| 1.37 | 3 | 0.01 | 7537 | 25 | 301 |
| 1.37 | 3 | 0.1 | 4924 | 25 | 189 |
| 1.37 | 3 | 1 | 1150 | 25 | 46 |
| 1.37 | 1 | 0.01 | 2600 | 25 | 104 |
| 1.37 | 1 | 0.1 | 2286 | 25 | 91 |
| 1.37 | 1 | 1 | 1051 | 25 | 41 |
| 1 | 3 | 0.01 | 1529 | 19 | 80 |
| 1 | 3 | 0.1 | 1645 | 19 | 87 |
| 1 | 3 | 1 | 547 | 19 | 29 |
| 1 | 1 | 0.01 | 480 | 19 | 25 |
| 1 | 1 | 0.1 | 480 | 19 | 25 |
| 1 | 1 | 1 | 442 | 19 | 23 |
is the relative refraction coefficient, is the slab thickness, is the absorption coefficient, and is the user time.