Ординатура / Офтальмология / Английские материалы / Handbook of Optical Coherence Tomography_Bouma, Tearney_2002

Figure 15 OCT images of impact-damaged epoxy/unidirectional E-glass composite

(a) 550 m from the surface along the xy plane and (b) along the xz plane, showing placement of tows via polyester stitching.

initiation and progression of damage is also planned. Imaging of short fiberglass thermoplastic composites is extremely important for determination of fiber orientation distribution, a critical property of this material in a high volume market. However, short fiberglass in thermoplastic composites is nominally 10 m in diameter and challenges the instrumental resolution. More far-reaching ideas use tunable sources to probe how moisture or other environmental fluids diffuse into the composite. Also, polarized OCT could be used to probe residual stress in composites, which could help in understanding their design and failure.

ACKNOWLEDGMENTS

I am grateful to Professor James Fujimoto for the use of his OCT instrumentation and insightful discussions. A number of members of his laboratory, including Drs. Brett Bouma, Jergen Herrmann and Xingde Li, and Mr. Rohit Prasankumar, kindly shared their time and expertise with us.

I also thank the people in my group, who contributed so much to the success of my work. I would first like to thank my group leader, Dr. Richard Parnas, for his full support of my work. I am grateful to Dr. Carl Zimba for much of the data collection

Figure 16 OCT images of impact-damaged epoxy/unidirectional E-glass composite (a) 652m from the surface along the xy plane and (b) along the xz plane, showing placement of tows via polyester stitching.

and for his willingness to mentor me. I also thank many other people who have contributed in their own way—Drs. Frederick Phelan for the modeling, Donald Hunston for his help on the damage aspects of the work, and Richard Peterson for permeability measurements. Special thanks to our technician, Ms. Kathleen Flynn, for doing all of the hard work.

Lastly, I thank Dr. Matt Everett and his group in the Medical Technology Program at Livermore National Laboratory for their continued efforts in helping me build an OCT system.

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16.2MODELS OF LIGHT PROPAGATION: MACROSCOPIC APPROACHES

Neglecting polarization, interference, and diffraction, the propagation of light through tissue can be modeled as neutral particle transport. At the macroscopic level, tissue can be characterized by three spatially varying, wavelength-dependent parameters [1]: the absorption coefficient ( a), the scattering coefficient ( s), and the scattering phase function ½pðs; s 0Þ&. a s denotes the probability of photon absorption for a s pathlength, s s denotes the probability of photon elastic scattering for a s pathlength, and the phase function is the probability density function that describes the likelihood of scattering from direction s to direction s 0 per unit solid angle. In a multiple scattering medium, photon particle transport can be described by the time-dependent Boltzmann equation [2],

ð1Þ

Here Lðr; s^Þ represents the local angular photon flux at position r at time t in unit direction s^, Sðr; s^Þ represents the source of photons generated at r and t in direction s^ and d 0 is an element of solid angle. This equation can be solved, subject to appropriate boundary conditions [2]; its solution can be used to estimate homogeneous tissue scattering and absorption properties from measurements of steady-state tissue reflectance and transmission [3].

The angular flux, Lðr; s^Þ, can be expanded as a sum of Legendre polynomials, and when scattering dominates absorption all but the first two terms in the expansion can be dropped. In this case, the transport equation reduces to the optical diffusion equation for the angle-independent photon flux, ðrÞ [4].

and represents the average cosine of the scattering angle. Closed-form analytical solutions can be obtained for the diffusion approximation to the radiative transport equation in both one and three dimensions for cases with a high degree of symmetry [2,4]. Alternatively, they can be solved numerically.

Another approach is to use Monte Carlo methods to simulate the random walk of photons within an absorbing and scattering tissue [5]. This approach offers a flexible, rigorous approach to describe photon transport in tissue according to the rules of radiative transport [6]. Monte Carlo based approaches are particularly useful for describing the propagation of light in heterogeneous tissues and close to boundaries. Welch and colleagues described a Monte Carlo technique to permit optical modeling that takes into account knowledge of complex three-dimensional anatomical structures such as small blood vessels [7]. Monte Carlo methods are used to

describe the propagation of photons through a tissue where the optical properties, stored in a material grid array, can vary arbitrarily in three dimensions [7].

Monte Carlo techniques have also been used to simulate the signal collected in OCT [8–10]. Pan et al. [8] described a Monte Carlo simulation of pathlength-resolved reflectance from multilayer tissues. This model assumed that all waveforms collected by the detector with a pathlength difference L less than the source coherence length will produce interference. Using this model, they predict that OCT signals are most sensitive to changes in index of refraction and least sensitive to changes in absorption coefficient over the range of optical properties found in tissue [8]. A similar approach was presented by Ducros [11]. Milner and colleagues extended this concept for OCT and optical Doppler tomography (ODT) [9,10]. Three-dimensional Monte Carlo models of accumulated photon pathlength were combined with a geometrical optics model of the probe geometry with low coherence interferometric detection [9]. They found that at depths of less than three mean free paths in highly scattering media, backscatter position of photons corresponded well with the focus position of the probe, but at greater depths localization of backscattering was lost due to detection of stray photons [9].

A drawback of Monte Carlo models is the computational time required to process the large number of photons required to yield good statistics. Several analytical models have been proposed to predict OCT signals in turbid media. The first analytical models [12–14] assumed that only light that has undergone a single backscattering event in the sample arm generates a heterodyne signal. However, light in the sample arm can be attenuated by scattering out of the detector collection path or by absorption events in the sample. The tissue optical properties incorporated in this model are the depth-dependent sample reflectivity, which is a function of the radar backscattering cross section of the scatterers in the sample volume ( b), and the extinction coefficient ( ext), which is related to the single-particle scattering cross section, the particle density, and the absorption coefficient. This model was shown experimentally to be accurate only at very small probing depths; beyond this, multiple scattering effects have an observable effect on the measured OCT signal [15,16].

Multiple scattering events detected in OCT are mostly forward-scattering events, which significantly degrade the signal due to beam spread and loss of spatial coherence. To incorporate these effects, the extended Huygens–Fresnel formulation of Yura [17] was adapted to OCT by Knuttel et al. [18] and then extended by Schmitt and Knuttel [17]. In this representation, loss of spatial coherence due to turbidity is characterized by the mutual coherence function (MCF) of the probing beam. The MCF is the cross-correlation of two optical, electric field vectors separated by a certain distance on a plane perpendicular to wave propagation. The lateral coherence length ( 0) is the characteristic separation at which the MCF falls to 1=e of its maximum value. Under certain conditions [19] the lateral coherence length can be expressed as

s

where sz characterizes the loss of coherence as the wave moves deeper into the medium, k ¼ 2= is the optical frequency, and rms is the mean scattering angle of

the scatterers in the tissue. All of these parameters depend on the scattering properties of the sample; these can be approximated using Mie theory if it is assumed that scattering is produced by a combination of individual spherical particles with a fractal size distribution as described in Ref. 20. With these assumptions, this model gives a reasonable approximation of the maximum tissue depth that can be probed with OCT.

In summary, analytical and computational models both indicate that OCT measurements are extremely sensitive to small changes in refractive index; a n as small as 0.01 can produce a significant interference modulation [8]. In most models of tissue optics, the index of refraction has been assumed to be homogeneous and nearly that of water and tissue scattering is characterized by a single scattering coefficient and anisotropy. However, the signal measured in OCT images (backscattering) is produced by local heterogeneities in refractive index. In order to understanding the microscopic and biological basis of OCT signals, tissue cannot be modeled as a homogeneous medium characterized simply by a scattering coefficient but must be considered an optically heterogeneous composite of microstructural segments with different refractive indices [8].

16.3MODELS OF LIGHT PROPAGATION: MICROSCOPIC APPROACHES

16.3.1 Background

To facilitate interpretation of data obtained using OCT or other scattering-based optical diagnostic techniques, mathematical models of light transport that account for refractive index fluctuations on a microscopic scale are needed. The initial step in developing such a model is to examine the interaction of light with a single cell, by considering the electromagnetic interaction of light with an arbitrarily heterogeneous biological cell. This approach allows investigation of how factors such as nuclear size and structure, organelle content, medium surrounding the cell, and incident light wavelength affect how light scatters from a cell.

Due to the size of scatterers in cells relative to the wavelengths used in optical imaging, electromagnetic methods are required to describe scattering. Mie theory has been used extensively to approximate scattering but generally requires modeling a cell as a homogeneous sphere. Schmitt and Kumar [21] presented a useful expansion of this idea by applying Mie theory to a volume of spheres with various sizes distributions. In addition, anomalous diffraction approximations [22], multiple solutions [23], and T-matrix computations [24] have been proposed and implemented. Each of these techniques offers significant advantages over conventional Mie theory; however, all require limiting geometrical and refractive index assumptions.

A more flexible approach is provided by a three-dimensional finite-difference time-domain (FDTD) model of cellular scattering [25]. Although computationally intensive, this model allows the computation of scattering patterns from inhomogeneous cells of arbitrary shape. The aim of the work described in this chapter is to develop an increased understanding of how light interacts with tissue on a cellular level using the FDTD model to predict cellular scattering patterns.

16.3.2 Origins of Cellular Scattering

Because scattering arises from mismatches in refractive index, when considering a cell from the perspective of how it will interact with light, the cell is viewed more appropriately as a continuum of refractive index fluctuations than as a single object containing a number of discrete particles. The magnitude and spatial extent of the index of refraction fluctuations arise from the physical composition and size of the components that make up the cell. Organelles and subcomponents of organelles having indices different from those of their surroundings are expected to be the primary sources of cellular scattering. The cell itself may be a significant source of small-angle scatter in applications such as flow cytometry in which cells are measured individually; however, for in vivo scattering-based diagnostics, the cell as an entity is not as important, because cells will be surrounded by other cells or tissue structures of similar index.

Certain organelles in cells are important potential sources of scattering. The nucleus is significant because it is often the largest organelle in the cell, and in diagnostic applications its size increases relative to the rest of the cell throughout neoplastic progression. Other potential scatterers include organelles whose size relative to the wavelength of light suggest that they may be important backscatterers. These include mitochondria (0.5–1:5 m), lysosomes (0:5 m), and peroxisomes (0:5 m). Mitochondria may be particularly influential in those cells that contain significant mitochondrial volume fractions because of the unique folded membrane structure of mitochondria. For instance, Chance and coworkers [26] found that mitochondria contribute 73% of the scattering from hepatocytes. Additionally, melanin, traditionally thought of as an absorber, must be considered an important scatterer owing to its size and high refractive index. Finally, structures consisting of membrane layers such as the endoplasmic reticulum or Golgi apparatus may prove significant because they will contain index fluctuations of high frequency and amplitude. Although the work presented here primarily concerns cells, to understand scattering from tissue, fibrous components such as collagen and elastin must be considered in addition to cellular matter. The relative importance of fibrous and cellular components depends upon tissue type.

16.3.3 Methods

Yee’s Method

Yee’s method [27] can be used to solve Maxwell’s curl equations using the finitedifference time-domain (FDTD) technique. The algorithm takes Maxwell’s curl equations and discretizes them in time and space, yielding six coupled finite-differ- ence equations. The six electric and magnetic field components (Ex; Ey; Ez; Hx; Hy; Hz) are spatially and temporally offset on a three-dimensional grid. The grid spacing must be less than =10 to yield accurate results. Except when otherwise noted, a =20 grid was used for the simulations presented here. As the six finite-difference equations are stepped in time, the electric and magnetic fields are updated for each grid point. To simulate propagation in an unbounded medium, boundary conditions must be applied to the tangential electric field components along the edges of the computational boundary at each time step. The Liao boundary condition [28] is used. The incident wave is a sinusoidal plane wave source.

The FDTD method computes the fields in a region around the cell that lies in the near field, which is then transformed to the far field. Parameters such as anisotropy and scattering cross section can be computed from the scattering pattern. The details of the FDTD model used in this work and the relevant calculations can be found in Ref. 25.

Simulation Parameters

The cell is constructed by assigning a permittivity value to each cell component. If desired, a range of permittivity values may be assigned to one component if that component, for example, the nucleus, is inhomogeneous. For a purely real refractive index, the dielectric constant is simply the square of the refractive index.

To accurately determine scattering patterns, it is necessary to model the cells in as physically realistic a manner as possible. Information about the size, quantity, and dielectric structure of cellular organelles was obtained from the literature and by examining phase contrast images and electron micrographs of cells and organelles of interest. Refractive index is a function of the concentration of macromolecules in a particular organelle. However, an organelle’s composition can vary significantly among different types of cells. For instance, it has been reported that the refractive index of the nucleus is higher than that of the cytoplasm in Chinese hamster ovary cells [29] and breast epithelial cells [30], whereas the cytoplasm was found to have a higher index than the nucleoid in E. coli [31]. Thus, accurate modeling of dielectric structure requires specific knowledge of organelle composition for the cell type of interest. Because this information is not readily available, refractive index values from the literature were assembled as a starting point for the simulations. The nucleus was always modeled with some inhomogeneities in refractive index. The distribution of index variations employed in the simulations was based on Fourier analysis of 100 phase contrast images of normal and cancerous human breast epithelial cells. The general ranges of refractive index values used in the FDTD simulations are shown in Table 1 [29,32–34].

16.3.4 Results

Verification

The simulation program was verified by computing the scattering patterns of homogeneous spheres ranging in diameter from 5 to 10 m and comparing the results to Mie theory. For small spheres, the two curves agree closely for all angles at horizontal and vertical polarizations For larger spheres, the FDTD pattern agrees very well for most angles but is somewhat greater than the Mie pattern for angles higher than 160 . The artificial increase is due to imperfect boundary conditions, resulting in artificial reflections at the edges of the computational domain. The scattering pattern of a 5 m sphere is shown in Fig. 1. The anisotropy parameter g and scattering cross section differ from theoretical values by less than 0.2% despite the artificial reflections.

Model output is presented in several ways: (1) scattering patterns, which illustrate the scattered intensity as a function of angle; (2) normalized scattering patterns, where the scattered power is normalized to the value at 0 ; and (3) scattering phase functions, Pð Þ, which present the scattered power versus angle, where the area under the curve has been normalized to 1. In the scattering pattern, it is visually easier to