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Text of the page (random words):
n 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 monte carlo simulation of a pencil beam incident on a semi infinite scattering medium biomedical applications of monte carlo methods edit biomedical imaging edit 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 edit 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 edit 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 edit 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 in our simplified model we use the following variance reduction technique to reduce computational time instead of propagating photons individually we create a photon packet with a specific weight generally initialized as unity as the photon interacts in the turbid medium it will deposit weight due to absorption and the remaining weight will be scattered to other parts of the medium any number of variables can be logged along the way depending on the interest of a particular application each photon packet will repeatedly undergo the following numbered steps until it is either terminated reflected or transmitted the process is diagrammed in the schematic to the right any number of photon packets can be launched and modeled until the resulting simulated measurements have the desired signal to noise ratio note that as monte carlo modeling is a statistical process involving random numbers we will be using the variable ξ throughout as a pseudo random number for many calculations schematic for modeling photon flow in an infinite scattering and absorbing medium with monte carlo simulations step 1 launching a photon packet edit in our model we are ignoring initial specular reflectance associated with entering a medium that is not refractive index matched with this in mind we simply need to set the initial position of the photon packet as well as the initial direction it is convenient to use a global coordinate system we will use three cartesian coordinates to determine position along with three direction cosines to determine the direction of propagation the initial start conditions will vary based on application however for a pencil beam initialized at the origin we can set the initial position and direction cosines as follows isotropic sources can easily be modeled by randomizing the initial direction of each packet x 0 position y 0 z 0 μ x 0 direction cosines μ y 0 μ z 1 displaystyle begin aligned x 0 text position y 0 z 0 mu _ x 0 text direction cosines mu _ y 0 mu _ z 1 end aligned step 2 step size selection and photon packet movement edit the step size s is the distance the photon packet travels between interaction sites there are a variety of methods for step size selection below is a basic form of photon step size selection derived using the inverse distribution method and the beer lambert law from which we use for our homogeneous model s ln ξ μ t displaystyle s frac ln xi mu _ t where ξ displaystyle xi is a random number and μ t displaystyle mu _ t is the total interaction coefficient i e the sum of the absorption and scattering coefficients once a step size is selected the photon packet is propagated by a distance s in a direction defined by the direction cosines this is easily accomplished by simply updating the coordinates as follows x x μ x s y y μ y s z z μ z s displaystyle begin aligned x leftarrow x mu _ x s y leftarrow y mu _ y s z leftarrow z mu _ z s end aligned step 3 absorption and scattering edit a portion of the photon weight is absorbed at each interaction site this fraction of the weight is determined as follows δ w μ a μ t w displaystyle delta w frac mu _ a mu _ t w where μ a displaystyle mu _ a is the absorption coefficient the weight fraction can then be recorded in an array if an absorption distribution is of interest for the particular study the weight of the photon packet must then be updated as follows w w δ w displaystyle w leftarrow w delta w following absorption the photon packet is scattered the weighted average of the cosine of the photon scattering angle is known as scattering anisotropy g which has a value between 1 and 1 if the optical anisotropy is 0 this generally indicates that the scattering is isotropic if g approaches a value of 1 this indicates that the scattering is primarily in the forward direction in order to determine the new direction of the photon packet and hence the photon direction cosines we need to know the scattering phase function often the henyey greenstein phase function is used then the scattering angle θ is determined using the following formula cos θ 1 2 g 1 g 2 1 g 2 1 g 2 g ξ 2 if g 0 1 2 ξ if g 0 displaystyle cos theta begin cases frac 1 2g left 1 g 2 left frac 1 g 2 1 g 2g xi right 2 right text if g neq 0 1 2 xi text if g 0 end cases and the polar angle φ is generally assumed to be uniformly distributed between 0 and 2 π displaystyle 2 pi based on this assumption we can set φ 2 π ξ displaystyle varphi 2 pi xi frac based on these angles and the original direction cosines we can find a new set of direction cosines the new propagation direction can be represented in the global coordinate system as follows μ x sin θ μ x μ z cos φ μ y sin φ 1 μ z 2 μ x cos θ μ y sin θ μ y μ z cos φ μ x sin φ 1 μ z 2 μ y cos θ μ z 1 μ z 2 sin θ cos φ μ z cos θ displaystyle begin aligned mu _ x frac sin theta mu _ x mu _ z cos varphi mu _ y sin varphi sqrt 1 mu _ z 2 mu _ x cos theta mu _ y frac sin theta mu _ y mu _ z cos varphi mu _ x sin varphi sqrt 1 mu _ z 2 mu _ y cos theta mu _ z sqrt 1 mu _ z 2 sin theta cos varphi mu _ z cos theta end aligned for a special case μ z 1 displaystyle begin aligned mu _ z 1 end aligned use μ x sin θ cos φ μ y sin θ sin φ μ z cos θ displaystyle begin aligned mu _ x sin theta cos varphi mu _ y sin theta sin varphi mu _ z cos theta end aligned or μ z 1 displaystyle begin aligned mu _ z 1 end aligned use μ x sin θ cos φ μ y sin θ sin φ μ z cos θ displaystyle begin aligned mu _ x sin theta cos varphi mu _ y sin theta sin varphi mu _ z cos theta end aligned c code indicatrix new direction cosines after scattering by angle theta fi mux new sin theta mux muz cos fi muy sin fi sqrt 1 muz 2 mux cos theta muy new sin theta muy muz cos fi mux sin fi sqrt 1 muz 2 muy cos theta muz new sqrt 1 muz 2 sin theta cos fi muz cos theta input muxs muys muzs direction cosine before collision mutheta fi cosine of polar angle and the azimuthal angle output muxd muyd muzd direction cosine after collision void indicatrix double muxs double muys double muzs double mutheta double fi double muxd double muyd double muzd double costheta mutheta double sintheta sqrt 1 0 costheta costheta sin theta double sinfi sin fi double cosfi cos fi if muzs 1 0 muxd sintheta cosfi muyd sintheta sinfi muzd costheta elseif muzs 1 0 muxd sintheta cosfi muyd sintheta sinfi muzd costheta else double denom sqrt 1 0 muzs muzs double muzcosfi muzs cosfi muxd sintheta muxs muzcosfi muys sinfi denom muxs costheta muyd sintheta muys muzcosfi muxs sinfi denom muys costheta muzd denom sintheta cosfi muzs costheta step 4 photon termination edit if a photon packet has experienced many interactions for most applications the weight left in the packet is of little consequence as a result it is necessary to determine a means for terminating photon packets of sufficiently small weight a simple method would use a threshold and if the weight of the photon packet is below the threshold the packet is considered dead the aforementioned method is limited as it does not conserve energy to keep total energy constant a russian roulette technique is often employed for photons below a certain weight threshold this technique uses a roulette constant m to determine whether or not the photon will survive the photon packet has one chance in m to survive in which case it will be given a new weight of mw where w is the initial weight this new weight on average conserves energy all other times the photon weight is set to 0 and the photon is terminated this is expressed mathematically below w m w ξ 1 m 0 ξ 1 m displaystyle w begin cases mw xi leq 1 m 0 xi 1 m end cases graphics processing units gpu and fast monte carlo simulations of photon transport edit monte carlo simulation of photon migration in turbid media is a highly parallelizable problem where a large number of photons are propagated independently but according to identical rules and different random number sequences the parallel nature of this special type of monte carlo simulation renders it highly suitable for execution on a graphics processing unit gpu the release of programmable gpus started such a development and since 2008 there have been a few reports on the use of gpu for high speed monte carlo simulation of photon migration 1 2 3 4 this basic approach can itself be parallelized by using multiple gpus linked together one example is the gpu cluster mcml which can be downloaded from the authors website monte carlo simulation of light transport in multi layered turbid media based on gpu clusters http bmp hust edu cn gpu_cluster gpu_cluster_mcml htm see also edit radiative transfer equation and diffusion theory for photon transport in biological tissue monte carlo method convolution for optical broad beam responses in scattering media monte carlo methods for electron transport links to other monte carlo resources edit optical imaging laboratory at washington university in st louis mcml oregon medical laser center photon migration monte carlo research at lund university sweden gpu acceleration of monte carlo simulations and scalable monte carlo open source code for download cloud based monte carlo for light transport in turbid scattering medium the tool is free to use in research and non commercial activities light transport in tissue as an example of monte carlo simulation with c source code references edit wang l h wu hsin i 2007 biomedical optics principles and imaging wiley l h wang s l jacques l q zhen...