4.1 Introduction
Optimising clinical PDT treatment regimes requires studies involving theoretical radiation transfer simulations. MCRT methods may be used to compute the light dose within tissue and assist in accurate light dosimetry for PDT treatment planning [1]. MCRT modelling is the most flexible approach to describing photon transport in biological tissue. However, any model must first be validated by experimental and theoretical comparison. In this chapter, the principles upon which the MCRT method operates will be explained followed by a description of the MCRT model used throughout this research. A light scattering experiment carried out for the purpose of experimentally validating the model will be described. In these experiments, lasers were directed through a cuvette loaded with varying concentrations of a scattering agent, known as Intralipid 20 %. Three lasers emitting at different wavelengths (405 nm, 532 nm and 632 nm) were used to investigate the scattering of each laser beam by the Intralipid 20 % and consequently the distribution of the light as it propagated through the scattering agent. Intralipid 20 % was used as it has been shown to be representative of the light scattering characteristics associated with tissue [2].
In addition, contours of fluence rate simulated by other MCRT models – previously published in the literature – were reproduced by our MCRT model for further validation. Firstly, the distribution of 476 nm light in human aorta with collimated incident beams of varying diameters simulated by Keijzer et al., [3] were compared to the light distributions simulated by the MCRT model. Secondly, Jacques et al., [4] generated contours representing the PDD administered to a tumour over a simulated treatment time and these were compared with additional MCRT simulations.
4.2 Operating Principles of the Monte Carlo Radiation Transfer Method The MCRT method was first put forward by Metropolis and Ulam in order to study physical processes using a statistical approach [5] and was later applied to the problem of light propagation in tissue by Wilson and Adam [6]. MCRT modelling is a technique that solves the transfer equation using the probabilistic nature of photon
interactions and has been used to simulate many such interactions, which have previously been modelled by several approximations, including the Kubelka-Munk theory [7, 8] and the diffusion approximation [9]. However, these techniques fail to accurately approximate light propagation at tissue surfaces, due to assumptions of isotropic scattering [10]. Conversely, MCRT modelling may be efficiently implemented for complicated tissue geometries and without restrictions in optical properties [11]. Furthermore, the MCRT technique provides accurate results for highly absorbing and scattering media at positions close to the surface and can handle the highly forward directed light scattering characteristics of tissue, i.e., anisotropic scattering [3, 12]. Wang et al. [13] developed a, now extensively used, MCRT model of light transport in multi-layered tissue for diagnostic and therapeutic applications of lasers in medicine. They modelled the tissue on a two dimensional (2D) grid system.
As previously shown in Chapter 1, Section 1.4, Equation 1.4, denotes the amount of radiant energy dEtransported across dA within a d over d during dt in units Joules, J. This total energy may be equally split among MCRT photons, which are essentially luminosity – energy – packets. Each energy packet, Ei is related to the specific intensity, I by the following equation
ˆ ( , , ) cos i E I r s t A t (4.1)
An energy packet, Ei may be expressed in units Joules,J, as
i
L t
E
N
(4.2)where L is the luminosity – energy per second – in Watts, t is the time in seconds and N is the number of MCRT photons.
MCRT photon energy packets are related to actual numbers of real photons, N , through the following equation
i i
E
N
h
(4.3)where h is Plank’s constant andi is the frequency of the MCRT photon energy packet.
The MCRT method is stochastic and statistical in nature, which describes the propagation of light by utilising probability distribution functions (PDFs). By sampling randomly from PDFs using cumulative distribution functions (CDFs), variables such as optical depths and photon scattering directions may be randomly chosen at interaction sites, enabling the position, direction and path of a photon to be determined. A computer generated pseudo-random number is used to represent a variable that is to be determined. The “random walk” of light photons may be determined as they traverse a tissue, until eventually getting absorbed or escape after traversing various distances. Consequently, photon path lengths between interactions are simulated. Variables are sampled randomly from the probability distribution function P x( ), which defines the distribution of the variable x over the interval
(a ≤ x ≤ b) as shown below
( )
1
b aP x dx
(4.4)A value of xi, where i = 1,2,3,…n is randomly generated many times based on a pseudo-random generator, which generates a random number, i,
( )
i x i aP x dx
(4.5)These equations are at the core of the MCRT method, which relies on random numbers. By implementing these equations it is possible to randomly sample the optical depth, , the albedo, a, and the scattering angles; cosine of the deflection angle, cos and the azimuthal angle, , which are established by random number generators. A photon travels a physical distance, S, before an interaction occurs and this is given by the optical depth, . The probability of travelling before an interaction is e. To randomly sample an optical depth, , the following applies
( )
P
e
(4.6)By correctly sampling many random , it is possible to get e after many samplings
0
e d
(4.7)Solving Equation 4.7 above yields
ln
(4.8)where is the optical depth and is a random number between [0,1]
By sampling random as described above, a photon pathlength, S – the distance to an interaction location – may then be computed as follows
0 S
t
ds
(4.9)where tis the total attenuation coefficient and S is the physical distance to an interaction site.
After a photon has travelled a randomly chosen , it is then either scattered or absorbed. The scattering probability is defined by the albedo, a, which can be expressed as s s a
a
(4.10)where s is the scattering coefficient and a is the absorption coefficient.
If a photon is absorbed, photon energy is deposited in the medium and can be re- emitted as fluorescence. The total attenuation coefficient, t sa. Typical values for a, s and t of the skin at 630 nm are 1.5 cm-1, 150 cm-1and 151.5 cm-1, respectively. For these parameters, an optical depth, 1, occurs at a depth of approximately 0.007 cm in the skin. A photon’s position on our code is given by three spatial Cartesian coordinates ( , , )x y z which are used to trace photon movement, and a photon’s direction of travel is described by three directional cosines