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# The Monte Carlo Method

## The Monte Carlo method

Monte Carlo calculation of photon transport is based on stochastic mathematical simulation of the interactions between photons and matter (for a review and general references on Monte Carlo techniques see, e.g., Andreo 1991). Photons are emitted (in a fictitious mathematical sense) from an isotropic point source into the solid angle specified by the focal distance and the x-ray field dimensions, and followed while they interact with the phantom according to the probability distributions of the physical processes that they may undergo: photo-electric absorption, coherent (Rayleigh) scattering or incoherent (Compton) scattering. This chain of interactions forms a so-called photon history. The cross sections for the photo-electric interaction, coherent scattering and incoherent scattering have been taken from Storm and Israel (1970) and the atomic form factors and incoherent scattering functions from Hubbell et al. (1975). Other interactions are not considered in PCXMC, because the maximum photon energy is limited to 150 keV. At these energies the range of secondary electrons in soft tissue is only a fraction of a millimetre, and the energy of the secondary electrons is approximated to be absorbed at the site of the photon interaction (except in calculating the bone marrow dose, see below). At each interaction point the energy deposition to the organ at that position is calculated and stored for dose calculation. A large number of independent photon histories is generated, and estimates of the mean values of the energy depositions in the various organs of the phantom are used for calculating the dose in these organs.

Pseudo random numbers are generated by a multiplicative linear congruential generator, MLCG(16807, 2^{31}-1) (Vattulainen et al. 1993), and are used for sampling the initial photon direction, distance between interactions, type of interacting atom, type of interaction and scattering angle (and the corresponding energy loss). To improve precision, the photons are constrained not to be absorbed by the photo-electric interaction; instead, the photo-electric absorption has been treated by associating a 'weight' to the photons. This weight, *w*, represents the expected proportion of photons that would have survived absorption in the preceding interactions, and is reduced in each interaction according to the probability of photo-electric absorption (*p*). At each interaction, an energy deposition of *w ^{.}p^{.}E*+

*w*(1-

^{.}*p*)

*is made to the organ where the interaction occurs, and the photon weight is reduced to*

**∆E**^{.}*w*

_{new}=

*w*

_{old}

**(1-**

^{.}*p*). Here,

*E*is the photon energy before the interaction and

*∆E*the energy loss in the scattering interaction. Each photon is followed until it exits the phantom without hitting it again, its energy falls to less than 2 keV (in which case it is forced to be absorbed), or until its weight is reduced to less than 0.003. In the last case, the photon is subjected to a game of Russian roulette: it is discarded with a probability of 0.75, but if it survives its weight is multiplied by a factor of four. Characteristic radiation resulting from the excitation of atoms in the body is not simulated, but it is assumed to be absorbed at the primary interaction site. There are no heavy elements in the phantoms; the maximum energy of such characteristic quanta would be about 4 keV and they would be absorbed near the primary interaction site.

The bones of the mathematical phantoms are modelled as a homogeneous mixture of mineral bone and organic constituents of the skeleton, including active bone marrow,. The overall composition of the skeleton is approximated as being constant over all bones in the body (see Table 4), but the amount of active bone marrow is varied from one part of the skeleton to another and is different for phantoms representing different ages (Cristy and Eckerman 1987). In reality, active bone marrow is located in small cavities in trabecular bone which causes the dose in the bone marrow to be higher than the kerma, due to secondary electrons from the bone matrix. This must be taken into account when calculating the dose to the active bone marrow. PCXMC calculates the dose in both components of the skeleton, the active bone marrow and the rest of skeletal material, by dividing the absorbed energy in the whole skeleton into two parts: active bone marrow and other constituents of bone (e.g., Rosenstein 1976a). For an energy deposition ∆*E* in a specific skeletal part *i* from a photon with energy *E*, the part of energy deposited in the active bone marrow in that skeleton part, ∆*E _{ABM, i}*

*,*is calculated by

where *m _{bone, i}* and

*m*denote the mass of the skeleton part

_{ABM, i}*i*, and the mass of active bone marrow in that skeleton part, respectively.

*µ*(

_{en}*E*)/

*rho*is the mass-energy absorption coefficient. The influence of the small cavity size on the dose in the active bone marrow is considered by multiplication with a photon energy dependent kerma-to-dose conversion factor (or dose enhancement factor),

*f*(

_{i}*E*), which increases the active bone marrow dose by a few percents when compared to the kerma (Kerr and Eckerman 1985, King and Spiers 1985). The size of the bone marrow cavities may vary depending on the age or the anatomical part of the skeleton (King and Spiers 1985), but this has not been taken into account in PCXMC: the same factor (Figure 3) is used for all bones and all phantom ages. The energy deposition in the other constituents of the skeleton (all other bone material than active bone marrow) is then

*∆E*-

*∆E*. The dose in the active bone marrow is calculated as a sum over all energy depositions and all parts of the skeleton according to Eq. (5), divided by the total mass of active bone marrow. For a discussion of various methods of calculating dose in the active bone marrow see the paper of Lee et al (2006b).

_{ABM, i}PCXMC calculates the organ doses for monochromatic photons of 10, 20, ..., 150 keV energy [or up to an user-defined maximum (which is below 150 keV)] in ten different batches of each energy value. This is sufficient, because the absorbed energy in any organ per photon is a smooth function of photon energy. The final estimate of the absorption at each energy value is obtained as the average of these batches, and the statistical error is estimated from the standard deviation of these batches. The doses and their statistical errors for a practical x-ray spectrum of interest are calculated afterwards by another module included in the program. The same Monte Carlo data can, therefore, be used for calculating doses for any spectrum of interest; such calculations are fast because they do not involve any further Monte Carlo simulations.

The x-ray spectra are calculated according to the theory of Birch and Marshal (1979) and are specified in terms of the x-ray tube voltage (kV), the angle of the tungsten target of the x-ray tube, and filtration. In the present version of the program, the user can simultaneously define two filters of arbitrary atomic number and thickness. The filter data are from the compiled x-ray interaction data of McMaster et al. (1969). Air kerma is calculated from photon fluence data using the conversion coefficients in ICRU (1992b) and Büermann et al (2006).

It should be noted that the accuracy of both the dose estimate and the estimate of its statistical error depend on the number of simulated interactions in the organ. This number may be low even for a large number of simulated photon histories if the dose in the organ is low or the organ small. It should also be noted that when the number of interactions is small, which is indicated by a high value of the statistical error, the estimate has a skewed non-normal distribution and the actual statistical errors may be higher than expected on the basis of the standard deviation.