Photon transport theories in Physics, Medicine, and Statistics (such as the Monte Carlo method), are commonly used to model light propagation in tissue.
Linearity indicates that a given response will increase by the same amount if the input is scaled and obeys the property of superposition.
Responses from photon transport methods can be physical quantities such as absorption, fluence, reflectance, or transmittance.
Given a specific physical quantity, G(x,y,z), from a pencil beam in Cartesian space and a collimated light source with beam profile S(x,y), a broad-beam response can be calculated using the following 2-D convolution formula: Similar to 1-D convolution, 2-D convolution is commutative between G and S with a change of variables
Because the inner integration of Equation 4 is independent of z, it only needs to be calculated once for all depths.
S0 is related to the total power P0 by Substituting Eq.
S0 is related to the total beam power P0 by Substituting Eq.
4, we obtain where First photon-tissue interactions always occur on the z axis and hence contribute to the specific absorption or related physical quantities as a Dirac delta function.
To convolve reliably for physical quantities at r in response to a top-hat beam, we must ensure that rmax in photon transport methods is large enough that r ≤ rmax − R holds.
For a Gaussian beam, no simple upper integration limits exist because it theoretically extends to infinity.
To calculate the response of a light beam on a plane perpendicular to the z axis, the beam function (represented by a b × b matrix) is convolved with the impulse response on that plane (represented by an a × a matrix).
The calculation efficiency of these two methods depends largely on b, the size of the light beam.
The calculation of each of these elements (except those near boundaries) includes b × b multiplications and b × b − 1 additions, so the time complexity is O[(a + b)2b2].