Jerome Spanier

Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues

We introduce a novel and efficient method to provide solutions to inverse photon migration problems in heterogeneous turbid media. The method extracts derivative information from a single Monte Carlo simulation to permit the rapid determination of rates of change in the detected photon signal with respect to perturbations in background tissue optical properties. We then feed this derivative information to a nonlinear optimization algorithm to determine the optical properties of the tissue heterogeneity under examination. We demonstrate the use of this approach to solve rapidly a two-region inverse problem of photon migration in the transport regime, for which diffusion-approximation-based approaches are not applicable.

Quasi-random methods for estimating integrals using relatively small samples

Much of the recent work dealing with quasi-random methods has been aimed at establishing the best possible asymptotic rates of convergence to zero of the error resulting when a finite-dimensional integral is replaced by a finite sum of integrand values. In contrast with this perspective to concentrate on asymptotic convergence rates, this paper emphasizes quasi-random methods that are effective for all sample sizes. Throughout the paper, the problem of estimating finite-dimensional integrals is used to illustrate the major ideas, although much of what is done applies equally to the problem of solving certain Fredholm integral equations. Some new techniques, based on error-reducing transformations of the integrand, are described that have been shown to be useful both in estimating high-dimensional integrals and in solving integral equations. These techniques illustrate the utility of carrying over to the quasi-Monte Carlo method certain devices that have proven to be very valuable in statistical (pseudoran...

Quasi-Monte Carlo Methods for Particle Transport Problems

Particle transport problems arise in such diverse application areas as the modeling of nuclear reactors and of semiconductor devices, and in the remote sensing of underground geologic features. Conventional Monte Carlo methods solve such problems by using pseudorandom numbers to make decisions at the microscopic level in order to draw conclusions about the macroscopic behavior of the system. Application of quasirandom (low discrepancy) sequences to such problems encounters certain difficulties that must be overcome if predictable gains over the use of pseudorandom Monte Carlo are to be realized. This paper outlines several ideas for achieving this and presents the results of “model” problem analyses and numerical tests of these ideas.

In vivo port-wine stain depth determination with a photoacoustic probe

We have designed a photoacoustic probe for port-wine stain (PWS) depth measurements consisting of optical fibers for laser light delivery and a piezoelectric element for acoustic detection. We characterized the capabilities and limitations of the probe for profiling PWS skin. The probe induced and measured photoacoustic waves in acrylamide tissue phantoms and PWS skin in vivo. The optical properties of the phantoms were chosen to mimic those of PWS skin. We denoised acoustic waves using spline wavelet transforms, then deconvolved with the impulse response of the probe to yield initial subsurface pressure distributions in phantoms and PWS skin. Using the phantoms, we determined that the limit in resolving epidermal and PWS layers was less than 70 microm. In addition, we used the phantoms to determine that the maximum epidermal melanin concentration that allowed detection of PWS was between 13 and 20%. In vivo measurements of PWS skin with different epidermal melanin concentrations correlated with the phantoms. Thus the photoacoustic probe can be used to determine PWS depth for most patients receiving laser therapy.

Post-Secondary Education

Since 1970 the number of U.S. college and university students choosing to major in mathematics has declined sharply. There are several sources of data describing this decline and many conjectures about the causes. The situation is summarized in two sections of the paper below.

Use of the δ-P 1 approximation for recovery of optical absorption, scattering, and asymmetry coefficients in turbid media

We introduce a robust method to recover optical absorption, reduced scattering, and single-scattering asymmetry coefficients (microa, micros, g1) of infinite turbid media over a range of (micros/microa) spanning 3 orders of magnitude. This is accomplished through the spatially resolved measurement of irradiance at source-detector separations spanning 0.25-8 transport mean free paths (l*). These measurements are rapidly processed by a multistaged nonlinear optimization algorithm in which the measured irradiances are compared with predictions given by the delta-P1 variant of the diffusion approximation to the Boltzmann transport equation. The ability of the delta-P1 model to accurately describe radiative transport within media of arbitrary albedo and on spatial scales comparable to l* is the key element enabling the separation of g1 from micros.

Analysis of single Monte Carlo methods for prediction of reflectance from turbid media

Starting from the radiative transport equation we derive the scaling relationships that enable a single Monte Carlo (MC) simulation to predict the spatially- and temporally-resolved reflectance from homogeneous semi-infinite media with arbitrary scattering and absorption coefficients. This derivation shows that a rigorous application of this single Monte Carlo (sMC) approach requires the rescaling to be done individually for each photon biography. We examine the accuracy of the sMC method when processing simulations on an individual photon basis and also demonstrate the use of adaptive binning and interpolation using non-uniform rational B-splines (NURBS) to achieve order of magnitude reductions in the relative error as compared to the use of uniform binning and linear interpolation. This improved implementation for sMC simulation serves as a fast and accurate solver to address both forward and inverse problems and is available for use at http://www.virtualphotonics.org/.

Efficient, automated Monte Carlo methods for radiation transport

Monte Carlo simulations provide an indispensible model for solving radiative transport problems, but their slow convergence inhibits their use as an everyday computational tool. In this paper, we present two new ideas for accelerating the convergence of Monte Carlo algorithms based upon an efficient algorithm that couples simulations of forward and adjoint transport equations. Forward random walks are first processed in stages, each using a fixed sample size, and information from stage k is used to alter the sampling and weighting procedure in stage k + 1. This produces rapid geometric convergence and accounts for dramatic gains in the efficiency of the forward computation. In case still greater accuracy is required in the forward solution, information from an adjoint simulation can be added to extend the geometric learning of the forward solution. The resulting new approach should find widespread use when fast, accurate simulations of the transport equation are needed.

Quantitative spectroscopy of superficial turbid media

We report a novel diffuse optical spectroscopy probe design for determining optical properties of superficial volumes of turbid samples. The fiber-based probe employs a highly scattering layer placed in contact with the sample of interest. This layer diffuses photons from a collimated light source before they enter the sample and provides a basis for describing light transported in superficial media by the diffusion approximation. We compare the performance of this modified two-layer diffusion model with Monte Carlo simulations. A set of experiments that demonstrate the feasibility of this method in turbid tissue phantoms is also presented. Optical properties deduced by this approach are in good agreement with those derived by use of a benchmark method for determining optical properties. The average interrogation depth of the probe design investigated here is estimated to be less than 1 mm.

Investigation of a probe design for facilitating the uses of the standard photon diffusion equation at short source-detector separations: Monte Carlo simulations

We design a special diffusing probe to investigate the optical properties of human skin in vivo. The special geometry of the probe enables a modified two-layer (MTL) diffusion model to precisely describe the photon transport even when the source-detector separation is shorter than 3 mean free paths. We provide a frequency domain comparison between the Monte Carlo model and the diffusion model in both the MTL geometry and conventional semiinfinite geometry. We show that using the Monte Carlo model as a benchmark method, the MTL diffusion theory performs better than the diffusion theory in the semiinfinite geometry. In addition, we carry out Monte Carlo simulations with the goal of investigating the dependence of the interrogation depth of this probe on several parameters including source-detector separation, sample optical properties, and properties of the diffusing high-scattering layer. From the simulations, we find that the optical properties of samples modulate the interrogation volume greatly, and the source-detector separation and the thickness of the diffusing layer are the two dominant probe parameters that impact the interrogation volume. Our simulation results provide design guidelines for a MTL geometry probe.