The Study of the Diffuse Equation About a Three-Layered Matched Medium (original) (raw)
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Applied Optics, 1997
The diffusion approximation of the radiative transfer equation is a model used widely to describe photon migration in highly diffusing media and is an important matter in biological tissue optics. An analysis of the time-dependent diffusion equation together with its solutions for the slab geometry and for a semi-infinite diffusing medium are reported. These solutions, presented for both the time-dependent and the continuous wave source, account for the refractive index mismatch between the turbid medium and the surrounding medium. The results have been compared with those obtained when different boundary conditions were assumed. The comparison has shown that the effect of the refractive index mismatch cannot be disregarded. This effect is particularly important for the transmittance. The discussion of results also provides an analysis of the role of the absorption coefficient in the expression of the diffusion coefficient.
Model for photon migration in turbid biological media
Journal of the Optical Society of America A, 1987
Various characteristics of photon diffusion in turbid biological media are examined. Applications include the interpretation of data acquired with laser Doppler blood-flow monitors and the design of protocols for therapeutic excitation of tissue chromophores. Incident radiation is assumed to be applied at an interface between a turbid tissue and a transparent medium, and the reemission of photons from that interface is analyzed. Making use of a discrete lattice model, we derive an expression for the joint probability r(n, p)d 2 p that a photon will be emitted in the infinitesimal area d 2 p centered at surface point p = (x, y), having made n collisions with the tissue. Mathematical expressions are obtained for the intensity distribution of diffuse surface emission, the probability of photon absorption in the interior as a function of depth, and the mean path length of detected photons as a function of the distance between the site of the incident radiation and the location of the detector. We show that the depth dependence of the distribution of photon absorption events can be inferred from measured parameters of the surface emission profile. Results of relevant computer simulations are presented, and illustrative experimental data are shown to be in accord with the theory.
1989
Abstruct-Using optical interaction coefficients typical of mammalian soft tissues in the red and near infrared regions of the spectrum, calculations of fluence-depth distributions, effective penetration depths and diffuse reflectance from two models of radiative transfer, diffusion theory, and Monte Carlo simulation are compared for a semi-infinite medium. The predictions from diffusion theory are shown to be increasingly inaccurate as the albedo tends to zero andlor the average cosine of scatter tends to unity.
Effect of the scattering delay on time-dependent photon migration in turbid media
Applied Optics, 1997
We modified the diffusion approximation of the time-dependent radiative transfer equation to account for a finite scattering delay time. Under the usual assumptions of the diffusion approximation, the effect of the scattering delay leads to a simple renormalization of the light velocity that appears in the diffusion equation. Accuracy of the model was evaluated by comparison with Monte Carlo simulations in the frequency domain for a semi-infinite geometry. A good agreement is demonstrated for both matched and mismatched boundary conditions when the distance from the source is sufficiently large. The modified diffusion model predicts that the neglect of the scattering delay when the optical properties of the turbid material are derived from normalized frequency-or time-domain measurements should result in an underestimation of the absorption coefficient and an overestimation of the transport coefficient. These observations are consistent with the published experimental data.
Equivalence of four Monte Carlo methods for photon migration in turbid media
Journal of the Optical Society of America A, 2012
In the field of photon migration in turbid media, different Monte Carlo methods are usually employed to solve the radiative transfer equation. We consider four different Monte Carlo methods, widely used in the field of tissue optics, that are based on four different ways to build photons' trajectories. We provide both theoretical arguments and numerical results showing the statistical equivalence of the four methods. In the numerical results we compare the temporal point spread functions calculated by the four methods for a wide range of the optical properties in the slab and semi-infinite medium geometry. The convergence of the methods is also briefly discussed.
Boundary conditions for the diffusion equation in radiative transfer
Journal of The Optical Society of America A-optics Image Science and Vision, 1994
Using the method of images, we examine the three boundary conditions commonly applied to the surface of a semi-infinite turbid medium. We find that the image-charge configurations of the partial-current and extrapolated-boundary conditions have the same dipole and quadrupole moments and that the two corresponding solutions to the diffusion equation are approximately equal. In the application of diffusion theory to frequency-domain photon-migration (FDPM) data, these two approaches yield values for the scattering and absorption coefficients that are equal to within 3%. Moreover, the two boundary conditions can be combined to yield a remarkably simple, accurate, and computationally fast method for extracting values for optical parameters from FDPM data. FDPM data were taken both at the surface and deep inside tissue phantoms, and the difference in data between the two geometries is striking. If one analyzes the surface data without accounting for the boundary, values deduced for the optical coefficients are in error by 50% or more. As expected, when aluminum foil was placed on the surface of a tissue phantom, phase and modulation data were closer to the results for an infinite-medium geometry. Raising the reflectivity of a tissue surface can, in principle, eliminate the effect of the boundary. However, we find that phase and modulation data are highly sensitive to the reflectivity in the range of 80-100%, and a minimum value of 98% is needed to mimic an infinite-medium geometry reliably. We conclude that noninvasive measurements of optically thick tissue require a rigorous treatment of the tissue boundary, boundary approach.
Fast perturbation Monte Carlo method for photon migration in heterogeneous turbid media
We present a two-step Monte Carlo (MC) method that is used to solve the radiative transfer equation in heterogeneous turbid media. The method exploits the one-to-one correspondence between the seed value of a random number generator and the sequence of random numbers. In the first step, a full MC simulation is run for the initial distribution of the optical properties and the " good " seeds (the ones leading to detected photons) are stored in an array. In the second step, we run a new MC simulation with only the good seeds stored in the first step, i.e., we propagate only detected photons. The effect of a change in the optical properties is calculated in a short time by using two scaling relationships. By this method we can increase the speed of a simulation up to a factor of 1300 in typical situations found in near-IR tissue spectroscopy and diffuse optical tomography, with a minimal requirement for hard disk space. Potential applications of this method for imaging of turbid media and the inverse problem are discussed.
arXiv (Cornell University), 2005
Does the diffusion coefficient of a photon depend on time ttt or the probability of absorption kkk? To find an answer to the question, photon transport in a medium of infinite extent is analyzed using the method of moments. It is pointed out that if DDD is defined so as to make it depend on ttt or kkk, it will also depend on the experimental conditions; that the parameter kkk which enters the stationary diffusion equation is in general different from that entering the transient version; and that a hitherto unused non-Markovian partial differential equation may be used for treating photon transport.
Applied Optics, 1992
When a picosecond light pulse is incident upon a turbid medium such as tissue, the temporal distribution of diffusely reflected and transmitted photons depends on the optical absorption and scattering properties of the medium. From diffusion theory it is possible to derive analytic expressions for the pulse shape in terms of the optical interaction coefficients of a homogeneous semi-infinite medium. Experimental tests of this simple model in tissue-simulating liquid phantoms of different geometries are presented here. The results of these tests show that, in a semi-infinite phantom, the application of the diffusion model provides estimates of the absorption and transport-scattering coefficients that are accurate to better than 10%. Comparable accuracy was also obtained with this simple model for finite slab, cylindrical, and spherical volumes as long as the objects were of sufficient size. For smaller volumes the absorption coefficient was overestimated because of the significant loss of photons at the boundaries of the object.