Research


Numerical modeling of light propagation in tissues and biomedical optical imaging

Near-infrared light probes noninvasively the interior of turbid media such as biological tissues. Of particular interest are medical applications where optical tomography and spectroscopy has the potential to provide diagnostic information about tumors in breast and prostate, and functional information about brain activities.

Near-infrared light is highly scattered in tissues and essentially "diffuses" inside the medium. Monte Carlo methods, versatile in modeling light propagation in such a medium, is inefficient and time prohibitive for many problems. The diffusion approximation to radiative transfer is simple but fails in early times and short distances. We developed a cumulant approximation to radiative transfer which models the correct behavior of photon migration at early times and reduces at long times to the center-moved diffusion approximation for transmission and back-scattering.

We have also investigated fast optical image reconstruction approaches (experimental setup and algorithm) with time-resolved ultrafast laser and regularization-inversion techniques. We developed time-sliced imaging, time-resolved Fourier optical diffuse tomography and three dimensional radiative transfer tomography of turbid media.

Nonlinear effects due to light multiple passage through one site was analyzed using self-energy diagram. A nonlinear correction factor (NCF) was then derived.

Another recent development is that we recognize independent component analysis (ICA) from information theory can be applied naturally to optical imaging. Absorptive, scattering, and fluorescent inhomogeneities can be localized and characterized by first separating independent components from the measurements. Small inhomogeneities were detected successfully within heterogeneous backgrounds.

The methods developed here have relevance in remote sensing and inverse problems in other fields.

Selected Publications

  1. M. Xu, M. Alrubaiee, S. K. Gayen, and R. R. Alfano. Three-dimensional optical tomographic imaging and localization of objects in turbid media using independent component analysis. Appl. Opt., 2004. (accepted).
  2. M. Xu, W. Cai, and R. R. Alfano. Multiple passages of light through an absorption inhomogeneity in optical imaging of turbid media. Opt. Lett., 29:1757-1759, 2004.
    [ .pdf | Abstract ]
  3. W. Cai, M. Xu, and R. R. Alfano. Three dimensional radiative transfer tomography for turbid media. IEEE JSTQE, 9:189-198, 2003.
    [ .pdf | Abstract ]
  4. M. Xu, W. Cai, M. Lax, and R. R. Alfano. Photon migration in turbid media using a cumulant approximation to radiative transfer. Phys. Rev. E, 65:066609, 2002.
    [ .pdf | Abstract ]
  5. M. Xu, W. Cai, M. Lax, and R. R. Alfano. A photon transport forward model for imaging in turbid media. Opt. Lett., 26(14):1066-1068, 2001.
    [ .pdf | Abstract ]
  6. M. Xu, M. Lax, and R. R. Alfano. Time-resolved Fourier optical diffuse tomography. J. Opt. Soc. Am. A, 18(7):1535-1542, 2001.
    [ .pdf | Abstract ]
  7. W. Cai, S. K. Gayen, M. Xu, M. Zevallos, M. Alrubaiee, M. Lax, and R. R. Alfano. Optical tomographic image reconstruction from ultrafast time-sliced transmission measurements. Appl. Opt., 38(19):4237-4246, 1999.
    [ .pdf | Abstract ]


Understanding light multiple scattering and propagation in turbid media

A deeper understanding of light propagation in turbid media is crucial to the success of optical imaging. We took a stochastic process point of view to study light propagation in turbid media. We analyzed how a pulse of light moves and expands in its propagation (cumulants of its distribution function). We used the cumulant approximation to obtain a better model than the diffusion approximation which is valid in both early and late time limits.

Recently, we investigated diffusion and depolarization of polarized light propagating in turbid media using a random walk model of vector photon. The second order statistics of the polarization and propagation directions of arbitrarily polarized light vs the number of scattering in the direction space is shown to be characterized by two eigenvalues for Mie scatterers of arbitrary size. Both light polarization and directionality anisotropies decay as $lambda_{+}^n$ when the number of scattering n is not too small. Light diffusion is found to be dependent on light polarization. Light diffusion in the initial linear polarization direction is hampened. Analytical expressions about the depolarization lengths of linear (or circular) polarized light are also derived. This length scale is proportional to the geometrical mean of the transport mean free path and the scattering mean free path.

Selected Publications

  1. M. Xu. Random walk of polarized light in turbid media. Phys. Rev. Lett., 2004. (to submit).
  2. M. Xu, W. Cai, M. Lax, and R. R. Alfano. Stochastic view of photon migration in turbid media. arXiv:cond-mat/0401409, 2004.
    [ http | Abstract ]
  3. M. Xu, W. Cai, M. Lax, and R. R. Alfano. Photon migration in turbid media using a cumulant approximation to radiative transfer. Phys. Rev. E, 65:066609, 2002.
    [ .pdf | Abstract ]
  4. W. Cai, M. Xu, M. Lax, and R. R. Alfano. Diffusion coefficient depends on time not on absorption. Opt. Lett., 27(9):731-733, 2002.
    [ .pdf ]


Anomalous light diffraction by statistics of geometrical path of rays, Mie scattering patterns, and particle sizing and shaping

We reformulate the anomalous-diffraction theory (ADT) of extinction of light by soft particles to be a statistical formalism about the geometrical paths of individual rays inside the particle. The dependence of light extinction on size, shape, and orientation of the particle is succinctly expressed by one probability function of the geometrical paths inside the particle. We analyzed the influence of particle shape on light extinction in this framework.

A simple approximation by assuming a Gaussian distribution of the geometrical paths in the particle has been shown intuitive and successful in modeling light extinction by biological cells and bacterias.

Selected Publications

  1. A. Katz, Alexandra Alimova, M. Xu, Paul Gottlieb, Elizabeth Rudolph, J. C. Steiner, and R. R. Alfano. In Situ determination of refractive index and size of Bacillus spores by light extinction. Opt. Lett., 2004. (accepted).
    [ Abstract ]
  2. A. Katz, A. Alimova, M. Xu, E. Rudolph, M. Shah, H. Savage, R. Rosen, S. A. McCormick, and R. R. Alfano. Bacteria size determination by elastic light scattering. IEEE JSTQE, 9:277-287, 2003.
    [ .pdf | Abstract ]
  3. M. Xu, M. Lax, and R. R. Alfano. Light anomalous diffraction using geometrical path statistics of rays and gaussian ray approximation. Opt. Lett, 28:179-181, 2003.
    [ .pdf | Abstract ]
  4. M. Xu. Light extinction and absorption by arbitrarily oriented finite circular cylinders using geometrical path statistics of rays. App. Opt., 42:6710-6723, 2003.
    [ .pdf | Abstract ]
  5. M. Xu and R. R. Alfano. More on patterns in Mie scattering. Opt. Comm., 226(1-6):1-5, 2003.
    [ .pdf | Abstract ]


Scientific computation

Expertise in Numeric Analysis, Modeling, Random Process, Monte Carlo Simulations, and Inverse Problems. I have independently developed codes including: a) Monte Carlo simulation of scalar and vector light propagation in turbid media, b) finite difference code solving light diffusion in slab, and c) various Tikhonov-type regularized inversion algorithms using L-curve and envelope guided conjugate gradients methods.

Scientific computing with C/C++, Ratfor/FORTRAN, and quick prototyping using Python and Matlab. Parallel computation using MPI. A note on the choice of a computational language is here.

Some old scientific calculation codes written by me can be found here.