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-matter interaction and the angles of deflection in a photon's trajectory when a scattering event occurs. This is equivalent to modeling photon transport analytically by the radiative transfer equation (RTE), which describes the motion of photons using a differential equation. However, closed-form solutions of the RTE are often not possible; for some geometries, the diffusion approximation can be used to simplify the RTE, although this, in turn, introduces many inaccuracies, especially near sources and boundaries. In contrast, Monte Carlo simulations can be made arbitrarily accurate by increasing the number of photons traced. For example, see the movie, where a Monte Carlo simulation of a pencil beam incident on a semi-infinite medium models both the initial ballistic photon flow and the later diffuse propagation.
The Monte Carlo method is necessarily statistical and therefore requires significant computation time to achieve precision. In addition Monte Carlo simulations can keep track of multiple physical quantities simultaneously, with any desired spatial and temporal resolution. This flexibility makes Monte Carlo modeling a powerful tool. Thus, while computationally inefficient, Monte Carlo methods are often considered the standard for simulated measurements of photon transport for many biomedical applications.
Biomedical applications of Monte Carlo methods
Biomedical imaging The optical properties of biological tissue offer an approach to biomedical imaging. There are many endogenous contrasts, including absorption from blood and melanin and scattering from nerve cells and cancer cell nuclei. In addition, fluorescent probes can be targeted to many different tissues. Microscopy techniques (including confocal, two-photon, and optical coherence tomography) have the ability to image these properties with high spatial resolution, but, since they rely on ballistic photons, their depth penetration is limited to a few millimeters. Imaging deeper into tissues, where photons have been multiply scattered, requires a deeper understanding of the statistical behavior of large numbers of photons in such an environment. Monte Carlo methods provide a flexible framework that has been used by different techniques to reconstruct optical properties deep within tissue. A brief introduction to a few of these techniques is presented here.
Photoacoustic tomography In PAT, diffuse laser light is absorbed which generates a local temperature rise. This local temperature variation in turn generates ultrasound waves via thermoelastic expansion which are detected via an ultrasonic transducer. In practice, a variety of setup parameters are varied (i.e. light wavelength, transducer numerical aperture) and as a result Monte Carlo modeling is a valuable tool for predicting tissue response prior to experimental methods. Diffuse optical tomography DOT is an imaging technique that uses an array of near-infrared light sources and detectors to measure optical properties of biological tissues. A variety of contrasts can be measured including the absorption due to oxy- and deoxy-hemoglobin (for functional neuro-imaging or cancer detection) and the concentration of fluorescent probes. In order to reconstruct an image, one must know the manner in which light traveled from a given source to a given detector and how the measurement depends on the distribution and changes in the optical properties (known as the forward model). Due to the highly scattering nature of biological tissue, such paths are complicated and the sensitivity functions are diffuse. The forward model is often generated using Monte Carlo methods.
Radiation therapy The goal of radiation therapy is to deliver energy, generally in the form of ionizing radiation, to cancerous tissue while sparing the surrounding normal tissue. Monte Carlo modeling is commonly employed in radiation therapy to determine the peripheral dose the patient will experience due to scattering, both from the patient tissue as well as scattering from collimation upstream in the linear accelerator.
Photodynamic therapy In Photodynamic therapy (PDT) light is used to activate chemotherapy agents. Due to the nature of PDT, it is useful to use Monte Carlo methods for modeling scattering and absorption in the tissue in order to ensure appropriate levels of light are delivered to activate chemotherapy agents.
Implementation of photon transport in a scattering medium Presented here is a model of a photon Monte Carlo method in a homogeneous infinite medium. The model is easily extended for multi-layered media, however. For an inhomogeneous medium, boundaries must be considered. In addition for a semi-infinite medium (in which photons are considered lost if they exit the top boundary), special consideration must be taken. For more information, please visit the links at the bottom of the page. We will solve the problem using an infinitely small point source (represented analytically as a Dirac delta function in space and time). Responses to arbitrary source geometries can be constructed using the method of Green's functions (or convolution, if enough spatial symmetry exists). The required parameters are the absorption coefficient, the scattering coefficient, and the scattering phase function. (If boundaries are considered the index of refraction for each medium must also be provided.) Time-resolved responses are found by keeping track of the total elapsed time of the photon's flight using the optical path length. Responses to sources with arbitrary time profiles can then be modeled through convolution in time.
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