Diffuse Photon Probes of Structural and Dynamical Properties of Turbid Media: Theory and Biomedical Applications

David A. Boas

A Dissertation in Physics

University of Pennsylvania


Feel free to contact me if you have any questions or comments. dboas@nmr.mgh.harvard.edu





Diffusing photons can be used to detect, localize, and characterize optical and dynamical spatial inhomogeneities embedded in turbid media. Measurements of the intensity of diffuse photons reveal information about the optical properties of a system. Speckle fluctuations carry information about the dynamical and optical properties. This dissertation shows that simple diffusion theories accurately model the intensity and speckle correlation signals that diffuse through turbid media with spatially varying properties and discusses possible biomedical applications.

We first look at the intensity of diffuse photons provided by a light source that is intensity modulated. This generates diffuse photon density waves (DPDW's) which exhibit classical wave behavior. We demonstrate experimentally and theoretically the refraction, diffraction, and scattering of DPDW's. Using accurate signal and noise models, we then present a detailed analysis which shows that DPDW's can be used to detect and locate objects larger than 3 mm and to characterize objects larger than 1 cm which are embedded inside turbid media with biologically relevant parameters. This diffuse photon probe should may find applications in medicine as a bed-side brain hematoma monitor, or for screening breast cancer, or other functional imaging applications.

We then consider the coherence properties of the diffuse photons as revealed by speckle intensity fluctuations and show that the temporal autocorrelation function of these fluctuations is accurately modeled by a correlation diffusion equation. Because the correlation diffusion equation is analogous to the photon diffusion equation, all concepts and ideas developed for DPDW's can be directly applied to the diffusion of correlation. We show experimentally and with Monte Carlo simulations that the diffusion of correlation can be viewed as a correlation wave that propagates spherically outwards from the source and scatters from macroscopic spatial variations in dynamical and/or optical properties. We also demonstrate the utility of inverse scattering algorithms for reconstructing images of the spatially varying dynamical properties of turbid media. The biomedical applicability of this diffuse probe is illustrated with examples of monitoring blood flow and probing the depth of burned tissue.



The potential to acquire information about tissue optical and dynamical properties non-invasively offers exciting possibilities for medical imaging. For this reason, the diffusion of near infrared photons (NIR) in turbid media has been the focus of substantial recent research. Applications range from pulse oximetry to tissue characterization to imaging of breast and brain tumors and to probing blood flow. Presently, pulse-time, amplitude modulated, and continuous wave sources of light are used to probe turbid media for optical anomalies such as tumors and hematomas.

These procedures are complicated by the fact that light does not travel ballistically through turbid media. Rather, photons experience many scattering events prior to their absorption or transmission through boundaries. For many biological tissues, the absorption length for NIR light is much longer than the scattering length. Furthermore, the scattering length is much smaller than the dimensions of the sample. In this case the migration of photons is accurately described as a diffusional process. These conditions are met in breast tissue for which the reduced scattering coefficient, m s' (which is the reciprocal of the photon random walk step), is approximately 10 cm-1 and the absorption coefficient, m a (which is the reciprocal of the photon absorption length), is approximately 0.03 cm-1.

An intensity modulated source of light produces a wave of light energy density which propagates spherically outwards from the source through the turbid medium. This intensity wave is called a diffuse photon density wave (DPDW). Although microscopically the photons are diffusing and have thus lost memory of their initial direction, macroscopically the photons combine incoherently to produce a scalar wave of light energy density with a well defined phase front. The wavelength of the DPDW depends on the optical properties and source modulation frequency and is around 10 cm for typical biological samples and modulation frequencies (~200 MHz). The optics of DPDW's have been well defined in the recent literature. In particular, studies of the distortion of DPDW's by optical inhomogeneities demonstrate that heterogeneities may be found and characterized by measuring distortions in the DPDW wavefront.

In a different vein, when a photon scatters from a moving particle, its frequency is Doppler-shifted by an amount that is proportional to the speed of the scattering particle and dependent on the scattering angle relative to the velocity of the scatterer. Under certain conditions it is possible to measure these small frequency shifts caused by Doppler scattering events. Thus it is possible to non-invasively measure particle motions and density fluctuations in a wide range of systems. Applications include measuring the Brownian motion of suspended macromolecules, velocimetry of flow fields, and in-vivo blood flow monitoring.

Methods for using light to measure flow and density fluctuations have appeared with numerous names including Photon Correlation Spectroscopy, Dynamic Light Scattering, Quasi-Elastic Light Scattering, and Diffusing Wave Spectroscopy. These methods basically fall into two categories: Doppler methods and speckle methods. The Doppler methods measure the Doppler broadening of the laser light linewidth directly using tunable optical filters. Speckle methods monitor the intensity fluctuations that arise from the beating of electric fields with slightly different frequencies. This is analogous to the acoustic beat notes that a musician uses to tune a musical instrument. The two different methods essentially give access to the same information, as is discussed by Briers.

Intensity and Doppler/speckle probes of random media are connected since they both rely on the behavior of the migrating photons. The two different probes require different equipment since the first measures the average intensity and the other indirectly measures the light coherence properties, but I show that the measured signals can be accurately predicted and quantified using analogous diffusion models.

Spectroscopic intensity probes have been used since the 1930's to measure blood oxygenation non-invasively and to detect hematomas and various breast cancers. Non-invasive monitoring of average blood oxygenation is successful and widely accepted. Detection of hematomas and cancers has also been successful but is not widely accepted because of the inability to accurately characterize the anomalies. To improve anomaly characterization it is necessary to have models which acc