In Chapter 9 (Section 9.5), we derived the diffusion equation for spatially uniform turbid media (Eq. (9.38)). If one is interested in measuring an average tissue property (say, the average hemoglobin concentration in muscle tissue, or the average hemoglobin saturation over a certain cerebral cortical volume), then Eq. (9.38) may provide a suitable analytical approach, within the limits of the assumption that the investigated tissue is spatially homogeneous. Of course, strictly speaking, this assumption is not correct, since biological tissues are not spatially homogeneous. At a microscopic level, it is the very presence of molecular species, intracellular organelles, and cellular structures that ultimately accounts for the absorption and scattering of light in tissues. However, these tissue inhomogeneities occur over spatial scales of microns (cellular structures), sub-microns (cellular organelles), and nanometers (structural proteins, organelle ultrastructure, biological macromolecules, etc.) that are not directly relevant to the macroscopic treatment of the absorption coefficient, diffusion coefficient, and optical energy density distribution that are considered by diffusion theory. By contrast, the presence of a macroscopic spatial heterogeneity in tissues, associated, say, with the skin and subcutaneous layers, bones, larger blood vessels, tendons, and multiple tissue types within the optically probed volume, may raise questions about the applicability of a model that assumes the spatial homogeneity of tissue.

In this chapter, we identify the length scale that is relevant in diffusion theory; in other words, the typical diffusion length that one should consider in comparison with the length scale over which the tissue optical properties vary. This chapter also derives expressions for the continuous-wave diffuse reflectance under a variety of conditions, and examines its dependence on the optical properties of the medium. Diffuse optical spectroscopy (principles described in Chapters 10–12 and experimental methods in Chapter 13) places an emphasis on the overall optical properties of the examined biological tissue, from which some relevant physiological or diagnostic parameters can be extracted. Diffuse optical imaging (described in Chapter 14), instead, places an emphasis on the spatial distribution of the optical properties of tissue at a macroscopic scale and is relevant in the detection of localized abnormalities (tumors, hemorrhagic or ischemic lesions, fluid-filled cysts, etc.) or focal events (targeted brain activation, localized hemodynamic responses, etc.). This chapter is devoted to continuous-wave methods for tissue spectroscopy, where the emission of the light source is constant; i.e., it is not time-varying.

Is a rose a rose? To crudely paraphrase Shakespeare, “What's in a name?” Can the same flower be described and characterized differently when viewed from different perspectives? Different scientific and technical fields tend to develop variations of the nomenclature for the same physical quantities. Within each field there is generally a degree of consistency, but interdisciplinary topics that bring together scientists, engineers, and practitioners from disparate fields often face inconsistencies in the symbols, units, and terminology utilized for quantitative analysis. The field of biomedical optics has not been immune to this problem, and publications in the field have invoked a variety of terms and symbols. In some cases, the same terms and symbols, as used in different fields, carry subtle but crucial differences in their meanings. An important example is the different meaning of the term intensity in physics (power per unit area), in radiometry (power per unit solid angle), and in heat transfer (power per unit area, per unit solid angle). Another example is that of the molar extinction coefficient for optical absorption of hemoglobin, which is typically defined using the logarithm to base-10 in chemistry and biology, or to base-e in physics, and may refer to one functional heme group or to the full molecule (four heme groups), depending on the specific physiological or biochemical characterizations. This may lead to a mismatch by as much as a factor of 9.2 (i.e., 4 × ln (10)) among the numerical values reported for the molar extinction coefficient of hemoglobin according to different conventions. For a long time, the three fields that most commonly make use of and describe (and teach) the methods of quantitative optical measurements have been physics, astronomy, and electrical engineering (or its subfield, optical engineering), although chemistry and biology also make use of optical characterization techniques. Because biomedical optics is a broad interdisciplinary field, which is based on contributions from researchers in a variety of specialty areas, it is important that a common language be used to describe and characterize its key quantitative parameters.

In this chapter, we define the nomenclature used in this book for the quantities that describe the optical radiation field and its interaction with biological tissue.

The word spectroscopy comes from the Latin-Greek hybrid, wherein spectrum is from the Latin word for “an appearance” (specter or apparition = a ghost!), and skopein is the Greek word for seeing or observing. Why does spectroscopy mean observing an appearance or a ghost? One explanation is based on the first systematic observation (among the many contributions to modern mathematics and science made by English physicist and mathematician Sir Isaac Newton [1642–1727]) of how the propagation of sunlight through a prism results in the “appearance” of a band of different colors. A second explanation may be that when one uses light to study an object, the observations are based on the effects of that object on light, so that one does not directly view the object, but one see its appearance, or a “ghost” of it. Finally, one can think of a spectral decomposition of sunlight as being filled with ghosts: the set of dark lines first observed by the English scientist William Wollaston [1766–1828] and the German physicist Joseph Fraunhofer [1787–1826], and the bands of invisible “colors” beyond the red on one side and beyond the violet on the other. The word spectrum has evolved in English to mean a range of ordered values or colors. For our purposes optical spectroscopy may be loosely defined as the study of atoms, molecules and particles by means of their wavelength-dependent interactions with light. Relevant to the methods discussed in this chapter light is broadly defined to encompass the range from the ultraviolet to mid-infrared wavelengths of the electromagnetic spectrum, or about 200 nm to 15 μm, which are transmitted through air with only moderate attenuation (see Figure 4.1). Over this range, the components used for optical instrumentation are most conveniently explained by appealing to the wave-like nature of light.

When it comes to scattering interactions and the diffusion of light in turbid media such as tissue, however, it is often more convenient to treat light as particles, or photons, rather than waves. Scattering can be elastic, in which the photon changes direction but not energy, or inelastic, in which the photon can both exchange energy and change its direction of propagation. Scattering phenomena are often described in probabilistic terms, by specifying the likelihood of a photon being scattered and the probable angular distribution of the scattering.

In Chapters 5, 6, and 7, we will examine in detail the molecular transitions and microscopic-scale structural features of tissue that are responsible for its optical properties on any scale, and that afford opportunities for diagnostic assessment. In this chapter, however, we introduce the primary constituents and structures of tissue that are responsible for its bulk optical properties, with primary attention to the wavelength range of near-UV through visible to near-IR. Of course, the heterogeneity of tissue lends complexity to the interpretation of the interactions of light with tissue, but, prior to addressing the effects of the heterogeneity, it is instructive to understand the factors that govern the averaged optical properties on a macroscopic scale.

The two optical properties that govern light transport in tissue, and that must be understood thoroughly, are absorption and scattering.

Other intrinsic optical properties of tissue that can affect light transport include birefringence and optical activity. These are subtle effects, however, that do not generally affect bulk transport properties of tissue, but can be relevant to microscopy based on phase contrast or differential interference contrast, which will be studied in Chapter 17 (Section 17.2).

Whether absorption or scattering dominates the propagation of light in tissue depends on the specifics of the tissue optical properties, which are generally a function of wavelength, and on their spatial distribution at microscopic and macroscopic levels. Figure 2.1 illustrates two cases, wherein either absorption or scattering dominates.

Absorption and scattering coefficients

We address the bulk optical properties, for which typical scale lengths are long compared to both the wavelength of light and the individual elements of tissue microstructure (cells, for example).

A textbook for a new field based on old concepts

Biomedical optics is a field that is both new and ancient. From the vantage point of the natural sciences and engineering, this is a newly developing interdisciplinary field, dealing with the application of optical science and technology to biological and biomedical problems, including clinical applications. On the other hand, the field has been around for thousands of years in a less quantitative way. Physicians’ eyes have served as optical spectrographs and sensors, with the brain serving as a database repository and providing the computational power (of a massively parallel computer) for pattern recognition. For example, physicians have known for a long time that a Caucasian patient with yellowing of the skin (or of the sclera of the eye) is likely to be suffering from liver disease. If the patient is flushed red, he/she might be running a fever, and if a local tissue area appears flushed and red, an inflammation is indicated; and the bluish appearance of a patient's lips and nail beds might be indicative of hypoxia. Now that the modern approach has become more quantitative and is developing new technologies, however, the field is growing and beginning to have a major impact on bioscience and healthcare. The emerging field combines the observational with the mathematical and computational, benefits from recent advances in optical technologies, and is coupled with a more rigorous physical-science approach that seeks to understand the basic underlying principles.

This textbook provides a broad survey of the field and covers the basics of a quantitative approach to the subtopics, taking advantage of the powerful tools offered by mathematics, physics and engineering. This quantitative approach and the didactic style, coupled with the description of representative applications and problem sets that accompany each chapter, are designed to serve the needs of students and professionals in engineering and the physical sciences. Students of the biological sciences will also find the text useful, especially if they have a good mathematical background. The basic material about general concepts and methods that are directly relevant to biomedical optics, including some topics of medical statistics, are described at an introductory level, whereas selected topics are covered in greater depth and treated at a more advanced level. Consequently, by proper selection of the material, this textbook can be used for upper-level undergraduate courses, as well as more advanced graduate-level courses on biomedical optics.

Chapter 7 introduced theoretical methods for calculating single scattering events of optical fields from individual particles, and illustrated examples of applications for measurement of individual cell properties by backscattering spectroscopy or by measurement of the scattering phase function. Chapters 9–12 will cover the concepts and theoretical framework for measurement of bulk tissue optical properties in the diffusion regime, where source-detector distances are large compared to 1/μ ′s (in most tissues, ≳0.5 cm), to enable application of the diffusion approximation to the radiative transport equation. Chapter 9, especially, rigorously addresses the parameter space for which the diffusion approximation to the transport equation is valid. In this chapter we cover the ground in between: the measurement of light that has scattered a few times, but not so many times that the light can be considered diffuse. The formalisms presented here, nonetheless, can be extended into the beginning of the diffusion regime, and may be advantageous where the P1 approximation holds (to be presented in Section 9.4). For the typical optical properties of tissue, this generally corresponds to distances less than 0.5 cm. Moreover, whereas the diffusion approximation also requires that the absorption coefficient be much lower than the reduced scattering coefficient, in the formalisms for short source-detector distances that are presented in this chapter that restriction is lifted. Consequently, the wavelengths typically used for short source-detector distances may span from the near-UV through the near infrared (NIR). In contrast, for applications in real tissue, the useful wavelength range for the diffusion regime is limited to the red-NIR wavelengths, where absorption by hemoglobin is minimal.

Given these considerations, it may seem confusing that the terms diffuse reflectance and diffuse reflectance spectroscopy (DRS) are commonly used to denote measurements of reflectance at any distance from the light source, including short distances. Such broad application of those terms may be partially justified by the fact that “diffuse” reflectance can be intended simply to distinguish between reflectance from a diffusing medium and single backscattering events or specular reflection (as from a mirror). DRS encompasses terms that more specifically relate to variations of the optical geometry and/or methods of the spectral analysis. These include elastic scattering spectroscopy (Bigio and Mourant, 1997), light scattering spectroscopy (Backman et al., 2001), differential pathlength spectroscopy (Amelink et al., 2004), and optical pharmacokinetics (Mourant et al., 1999) among others.