Richard C. Haskell

BOUNDARY-CONDITIONS FOR THE DIFFUSION EQUATION IN RADIATIVE-TRANSFER

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, and we suggest a unified partial-current--extrapolated boundary approach.

Properties of photon density waves in multiple-scattering media

Amplitude-modulated light launched into multiple-scattering media, e.g., tissue, results in the propagation of density waves of diffuse photons. Photon density wave characteristics in turn depend on modulation frequency (omega) and media optical properties. The damped spherical wave solutions to the homogeneous form of the diffusion equation suggest two distinct regimes of behavior: (1) a high-frequency dispersion regime where density wave phase velocity V(p) has a radicalomega dependence and (2) a low-frequency domain where V(p), is frequency independent. Optical properties are determined for various tissue phantoms by fitting the recorded phase (?) and modulation (m) response to simple relations for theappropriate regime. Our results indicate that reliable estimates of tissue like optical properties can be obtained, particularly when multiple modulation frequencies are employed.

An optical coherence microscope for 3-dimensional imaging in developmental biology

An optical coherence microscope (OCM) has been designed and constructed to acquire 3-dimensional images of highly scattering biological tissue. Volume-rendering software is used to enhance 3-D visualization of the data sets. Lateral resolution of the OCM is 5 mm (FWHM), and the depth resolution is 10 mm (FWHM) in tissue. The design trade-offs for a 3-D OCM are discussed, and the fundamental photon noise limitation is measured and compared with theory. A rotating 3-D image of a frog embryo is presented to illustrate the capabilities of the instrument.

PORTABLE, HIGH-BANDWIDTH FREQUENCY-DOMAIN PHOTON MIGRATION INSTRUMENT FOR TISSUE SPECTROSCOPY

We describe a novel frequency-domain photon migration instrument employing direct diode laser modulation and avalanche photodiode detection, which is capable of noninvasively determinating the optical properties of biological tissues in near real time. An infinite medium diffusion model was used to extract absorption and transport scattering coefficients from 300-kHz to 800-MHz photon-density wave phase data. Optical properties measured in tissue-simulating solutions at 670 nm agreed to within 10% of those expected.

Tissue Characterization and Imaging Using Photon Density Waves

The optical properties of brain tissues have been evaluated by measuring the phase velocity and attenuation of harmonically modulated light. The phase velocity for photon density waves at 650-nm wavelength has been found to be in the range of 5 to 12% of the corresponding velocity in a nonscattering medium, and the optical penetration depth was in the range 2.9 to 5.2 mm. These results are used to predictthe resolution of optical imaging of deep tissue structures by diffusely propagating incoherent photons. The results indicate that structures of a few millimeters in linear dimension can be identified at 10 mm depth provided that proper wavelength and time resolution are selected. This depth can possibly be enlarged to 30 mm in the case of tissues with very low scattering such as in the case of the neonatal human brain.

Form birefringence of muscle

We investigate the sensitivity of measurements of muscle birefringence to cross-bridge dynamics in the resting, active, and rigor states. The theory of form birefringence is reviewed, and an optical model is constructed for the form birefringence of muscle. Values for the parameters in the model are selected or deduced from the literature. As an illustration of the use of the model, plausible distributions for the orientations of cross-bridges in the resting, active, and rigor states are constructed using a model for cross-bridge dynamics suggested by Huxley and Kress (1985). The general magnitude of the predictions of our model is comparable with that of published measurements of muscle birefringence. However, the precise values of the predicted birefringence for the resting, active, and rigor states are sensitive to the assumed orientations of cross-bridges. We also investigate the dependence of muscle birefringence on sarcomere length and on disorder in the orientation of the myofilament array. We conclude that measurements of muscle birefringence can play a useful role in distinguishing between proposed models of cross-bridge dynamics.

Office-based optical coherence tomographic imaging of human vocal cords.

Optical coherence tomography (OCT) is an evolving noninvasive imaging modality and has been used to image the larynx during surgical endoscopy. The design of an OCT sampling device capable of capturing images of the human larynx during a typical office based laryngoscopy examination is discussed. Both patients and physicians movements were addressed. In vivo OCT imaging of the human larynx is demonstrated. Though the long focal length limits the lateral resolution of the image, the basement membrane can still be readily distinguished. Office-based OCT has the potential to guide surgical biopsies, direct therapy, and monitor disease. This is a promising imaging modality to study the larynx.

Determination of the optical properties of the human uterus using frequency-domain photon migration and steady-state techniques

The optical properties (absorption and transport scattering coefficients) of freshly excised, bulk human uterine tissues were measured at 630 nm using frequency-domain and steady-state photon migration techniques. Measurements were made on both normal (pre- and post-menopausal) and non-neoplastic fibrotic tissues. The absorption coefficient of normal post-menopausal tissue (approximately 0.06 mm(-1)) was found to be significantly greater than that of normal pre-menopausal tissue (0.02-0.03 mm(-1)) and pre-menopausal fibrotic tissue (0.008 mm(-1)). The transport scattering coefficient was similar in all three tissue types considered (0.6-0.9 mm(-1)). From the preliminary results presented here, we conclude that optical properties can be reliably calculated either from the frequency-dependent behaviour of diffusely propagating photon density waves or by combining the frequency-independent photon density wave phase velocity with steady-state light penetration depth measurements. Instrument bandwidth and tissue absorption relaxation time ultimately determine the useful frequency range necessary for frequency-domain photon migration (FDPM) measurements. Based on the optical properties measured in this study, we estimate that non-invasive FDPM measurements of normal uterine tissue require modulation frequencies in excess of 350 MHz.

Phase Modulation at 125 kHz in a Michelson Interferometer Using an Inexpensive Piezoelectric Stack Driven at Resonance

Fast phase modulation has been achieved in a Michelson interferometer by attaching a lightweight reference mirror to a piezoelectric stack and driving the stack at a resonance frequency of about 125 kHz. The electrical behavior of the piezo stack and the mechanical properties of the piezo-mirror arrangement are described. A displacement amplitude at resonance of about 350 nm was achieved using a standard function generator. Phase drift in the interferometer and piezo wobble were readily circumvented. This approach to phase modulation is less expensive by a factor of roughly 50 than one based on an electro-optic effect.

Quasi-elastic light-scattering studies of single skeletal muscle fibers

Measurements were made of the intensity autocorrelation function, g(2)[tau], of light scattered from intact frog muscle fibers. During the tension plateau of an isometric tenanus, scattered field statistics were approximately Gaussian and intensity fluctuations were quasi-stationary. The half time, tau 1/2, for the decay of g(2)[tau] was typically 70 ms at a scattering angle of 30 degrees. The decay rate, 1/tau 1/2, of g(2)[tau] varied roughly linearly with the projection of the scattering vector on the fiber axis. 1/tau 1/2 was greater during the tension creep phase of tetani of highly stretched fibers, but was roughly independent of sarcomere length during the tension plateau. g(2)[tau] measured during rest or on diffraction pattern maxima during isometric contraction were flat with low amplitudes. These results are consistent with a model of a 200-mu m segment of an isometrically contracting fiber in which scattering material possesses relative axial velocities of 1-2 mu m/s accompanied by relative axial displacements greater than 0.1 mu m. The slow (1-2 mu m/s) motion of one portion of the fiber relative to another observed under the microscope (500X) during isometric contraction is consistent with the light-scattering results. Structural fluctuations on the scale of the myofibrillar sarcomere which may arise from asynchronous cycling of cross-bridges must involve relative axial velocities less than 3 mu m/s or relative axial displacements less than 0.05 mu m.