Preface to the Second Edition

The choice of the approach is dictated by both the structural specificity of the tissue under study and the kind of light scattering characteristics that are to be obtained.

Most tissues are composed of structures with a wide range of sizes, and most

can be described as a random continuum of inhomogeneities of the refractive index with a varying spatial scale.154,155 Phase contrast microscopy has been used

in particular to show that the structure of the refraction index inhomogeneities in mammalian tissues is similar to the structure of frozen turbulence in a number of cases.154 This fact is of fundamental importance for understanding the peculiarities of light propagation in tissue, and it may be a key to the solution of the inverse problem of tissue structure reconstruction. This approach is applicable for tissues with no pronounced boundaries between elements that feature significant

heterogeneity. The process of scattering in these structures may be described under certain conditions using the model of a phase screen.75,136,155,157

The second approach to tissue modeling is its representation as a system of discrete scattering particles. In particular, this model has been advantageously used

to describe the angular dependence of the polarization characteristics of scattered radiation.145,146,148,150,158 Blood is the most important biological example of a disperse system that entirely corresponds to the model of discrete particles.48,101,159

Biological media are often modeled as ensembles of homogeneous spherical particles, since many cells and microorganisms, particularly blood cells, are close in shape to spheres or ellipsoids. A system of noninteracting spherical particles is

the simplest tissue model. Mie theory rigorously describes the diffraction of light in a spherical particle.148,160 The development of this model involves taking into

account the structures of the spherical particles, namely, the multilayered spheres and the spheres with radial nonhomogeneity, anisotropy, and optical activity.145,146

Because connective tissue consists of fiber structures, a system of long cylinders is the most appropriate model for it. Muscular tissue, skin dermis, dura mater, eye cornea, and sclera belong to this type of tissue formed essentially by collagen fibrils. The solution of the problem of light diffraction in a single homogeneous or multilayered cylinder is also well understood.148

The sizes of cells and tissue structure elements vary in size from a few tenths

of nanometers to hundreds of micrometers.47,58,94–96,129,130,135,138,149–153,161–180

Blood cells (erythrocytes, leukocytes, and platelets) exhibit the following parameters. A normal erythrocyte in plasma has the shape of a concave-concave disk with a diameter varying from 7.1 to 9.2 μm, a thickness of 0.9–1.2 μm in the center and 1.7–2.4 μm on the periphery, and a volume of 90 μm3. Leukocytes are formed like spheres with a diameter of 8–22 μm. Platelets in the bloodstream are biconvex disklike particles with diameters ranging from 2 to 4 μm. Normally, blood has about 10 times as many erythrocytes as platelets and about 30 times as many platelets as leukocytes.

Most other mammalian cells have diameters in the range of 5–75 μm. In the epidermal layer, the cells are large (with an average cross-sectional area of about 80 μm2) and quite uniform in size. Fat cells, each containing a single lipid droplet that nearly fills the entire cell and therefore results in eccentric placement of the

cytoplasm and nucleus, have a wide range of diameters, from a few microns to 50–75 μm. Fat cells may reach diameters of 100–200 μm in pathological cases.

There are a wide variety of structures within cells that determine tissue light scattering (see Fig. 1.1). Cell nuclei are on the order of 5–10 μm in diameter; mitochondria, lysosomes, and peroxisomes have dimensions of 1–2 μm; ribosomes are on the order of 20 nm in diameter; and structures within various organelles can have dimensions of up to a few hundred nanometers. Usually, the scatterers in cells are not spherical. The models of prolate ellipsoids with a ratio of the ellipsoid axes between 2 and 10 are more typical.

Figure 1.1 Major organelles and inclusions of the cell.129

The hollow organs of the body are lined with a thin, highly cellular surface layer of epithelial tissue, which is supported by underlying, relatively acellular connective tissue. In healthy tissues, the epithelium often consists of a single wellorganized layer of cells with en face diameter of 10–20 μm and height of 25 μm (see Fig. 1.2). In dysplastic epithelium, cells proliferate and their nuclei enlarge and appear darker (hyperchromatic) when stained.150 Enlarged nuclei are primary indicators of cancer, dysplasia, and cell regeneration in most human tissues.

In fibrous tissues or tissues containing fiber layers (cornea, sclera, dura mater, muscle, myocardium, tendon, cartilage, vessel wall, retinal nerve fiber layer, etc.) and composed mostly of microfibrils and/or microtubules, typical diameters of the cylindrical structural elements are 10–400 nm. Their length is in a range from 10– 25 μm to a few millimeters.

Figure 1.2 Microphotograph of the isolated normal intestinal epithelial cells (a) and intestinal malignant cell line T84 (b). Note the uniform nuclear size distribution of the normal epithelial cell (a) in contrast to the T84 malignant cell line, which at the same magnification shows larger nuclei and more variation in nuclear size (b). Solid bars equal 20 μm in each panel (from Ref. 150 © 1999 IEEE).

The dominant scatterers in an artery may be the fibers, cells, or subcellular organelles. Muscular arteries have three main layers. The inner intimal layer consists of endothelial cells with a mean diameter of less than 10 μm. The medial layer consists mostly of closely packed smooth muscle cells with a mean diameter of 15–20 μm; small amounts of connective tissue, including elastin, collagenous, and reticular fibers, as well as a few fibroblasts, are also located in the medial. The outer adventitial layer consists of dense fibrous connective tissue that is largely made up of 1- to 12-μm-diameter collagen fibers and thinner, 2- to 3-μm-diameter elastin fibers.

Another two examples of complex scattering structures are the myocardium and the retinal nerve fiber layer. The myocardium consists mostly of cardiac muscle, which is comprised of myofibrils (about 1 μm in diameter) that in turn consist of cylindrical myofilaments (6–15 nm in diameter) and aspherical mitochondria (1–2 μm in diameter). The retinal nerve fiber layer comprises bundles of unmyelinated axons that run across the surface of the retina. The cylindrical organelles of the retinal nerve fiber layer are axonal membranes, microtubules, neurofilaments, and mitochondria. Axonal membranes, like all cell membranes, are thin (6–10 nm) phospholipid bilayers that form cylindrical shells enclosing the axonal cytoplasm. Axonal microtubules are long tubular polymers of the protein tubulin with an outer diameter of ≈25 nm, an inner diameter ≈15 nm, and a length of 10–25 μm. Neurofilaments are stable protein polymers with a diameter ≈10 nm. Mitochondria are ellipsoidal organelles that contain densely involved membranes of lipid and protein. They are 0.1–0.2 μm thick and 1–2 μm long.

For some tissues, the size distribution of the scattering particles may be essentially monodispersive, and for others it may be quite broad. Two opposing

examples are a transparent eye cornea stroma, which has a sharply monodisper-

sive distribution, and a turbid eye sclera, which has a rather broad distribution of collagen fiber diameters.129,130 There is no universal distribution size func-

tion that would describe all tissues with equal adequacy. In optics of dispersed systems, Gaussian, gamma, or power size distributions are typical.171 Polydispersion for randomly distributed scatterers can be accounted for by using the gamma-

distribution or the skewed logarithmic distribution of scatterers’ diameters, cross sections, or volumes.61,129,154,156,165,172 In particular, for turbid tissues such as eye

sclera, the gamma radii distribution function is applicable.61,172

Absorbed light is converted to heat or radiated in the form of fluorescence; it is also consumed in photobiochemical reactions. The absorption spectrum depends on the type of predominant absorption centers and water content of tissues

(see Figs. 1.3–1.7). Absolute values of absorption coefficients for typical tissues

lie in the range 10−2 to 104 cm−1.1–4,6,9–15,28,29,31,37–42,56,57,72,86–91 In the ultravio-

let (UV) and infrared (IR) (λ ≥ 2000 nm) spectral regions, light is readily absorbed, which accounts for the small contribution of scattering and the inability of radiation to penetrate deep into tissues (only through one or two cell layers). Short-wave visible light penetrates typical tissues as deep as 0.5–2.5 mm, whereupon it undergoes an e-fold decrease of intensity. In this case, both scattering and absorption occur, with 15–40% of the incident radiation being reflected. In the 600–1600-nm wavelength range, scattering prevails over absorption, and light penetrates to a depth of 8–10 mm. Simultaneously, the intensity of the reflected radiation increases to 35–70% of the total incident light (due to backscattering).

Figure 1.3 The absorption spectrum of water.56

Light interaction with a multilayer and multicomponent skin is a very complicated process.57 The horny-skin layer (stratum corneum) reflects about 5–7% of the incident light. A collimated light beam is transformed to a diffuse one by microscopic inhomogeneities at the air/horny-layer interface. A major part of reflected light results from backscattering in different skin layers (stratum corneum,

Figure 1.4 Molar attenuation spectra for solutions of major visible light-absorbing human skin pigments: 1, DOPA-melanin (H2O); 2, oxyhemoglobin (H2O); 3, hemoglobin (H2O); 4, bilirubin (CHCl3).57

Figure 1.5 The transmittance spectrum of a 3-mm-thick slab of female breast tissue. A spectrometer with an integrating sphere was used. The contributions of absorption bands of the tissue components are marked: 1, hemoglobin; 2, fat; and 3, water.50

epidermis, dermis, blood, and fat). The absorption of diffuse light by skin pigments is a measure of bilirubin content, hemoglobin concentration, and its saturation with oxygen, and the concentration of pharmaceutical products in blood and tissues; these characteristics are widely used in the diagnosis of various diseases (see Fig. 1.4). Certain phototherapeutic and diagnostic modalities take advantage of ready transdermal penetration of visible and near-infrared (NIR) light inside the body in the wavelength region, corresponding to the therapeutic or diagnostic window (600–1600 nm) (Fig. 1.7).

Another example of heterogeneous multicomponent tissue is a female breast (which is principally composed of adipose and fibrous tissues). The absorption bands of hemoglobin, fat, and water are clearly seen in vitro in the measured spec-

700 and 1100 nm, and narrow ones at about 1300 and 1600 nm, where the lowest percentage of light is attenuated.

Solid tissues such as ribs and the skull, as well as whole blood, are also easily penetrable by visible and NIR light.1–4,6,9–16,36,91,129,130 The relatively good

transparency of skin for long-wave UV light (UVA) depends on DNA, tryptophane, tyrosine, urocanic acid, and melanin absorption spectra and underlies se-

lected methods of photochemotherapy of skin tissues using UVA irradiation (see

Fig. 1.4).3,6,10,57,86,129,130

A collimated (laser) beam is attenuated in a thin tissue layer of thickness d in accordance with the Bouguer-Beer-Lambert exponential law as37

where I (d) is the intensity of transmitted light measured using a distant photodetector with a small aperture (on-line or collimated transmittance), W/cm2; RF is the coefficient of Fresnel reflection; at the normal beam incidence, RF = [(n − 1)/(n + 1)]2; n is the relative mean refractive index of tissue and surrounding media; I0 is the incident light intensity, W/cm2;

is the extinction coefficient (interaction or total attenuation coefficient), 1/cm, where μa is the absorption coefficient, 1/cm, and μs is the scattering coefficient, 1/cm. Strictly speaking, Eq. (1.1) is valid only for a highly absorbing media, when μa μs.

The extinction coefficient is connected with the extinction cross section σext as

where ρs is the density of particles (tissue and cell compounds). For a system of particles with absorption,

The average scattering cross section per particle can be presented in a form suitable for experimental evaluations:148

where I0 is the intensity of the incident light, I (θ) is the angular distribution of the scattered light by a particle, and θ is the scattering angle. For macroscopically

isotropic and symmetric media, the average scattering cross section is independent of the direction and polarization of the incident light. The average extinction, σext, and absorption, σabs, cross sections are also independent of the direction and polarization state of the incident light.

The probability that a photon incident on a small volume element will survive is equal to the ratio of the scattering and extinction cross sections, and is called the “albedo” for single scattering, :

The albedo ranges from zero for a completely absorbing medium to unity for a completely scattering medium.

The mean free path length (MFP) between two interactions is denoted by

1.1.2 Theoretical description

To analyze light propagation under multiple scattering conditions, it is assumed that absorbing and scattering centers are uniformly distributed across the tissue. UV-A, visible, or NIR radiation is normally subject to anisotropic scattering characterized by a clearly apparent direction of photons undergoing single scattering,

which may be due to the presence of large cellular organelles [mitochondria, lysosomes, and inner membranes (Golgi apparatus)].3,58,85,95,96,129,130,135,150–153

When the scattering medium is illuminated by unpolarized light and/or only the intensity of multiply scattered light needs to be computed, a sufficiently strict mathematical description of continuous wave (CW) light propagation in a medium

is possible in the framework of the scalar stationary radiation transfer theory

(RTT).1,3,6,12–16,129,130,135,136,145,146,181–197

This theory is valid for an ensemble of scatterers located far from one another and has been successfully used to work out some practical aspects of tissue optics. The main stationary equation of RTT for monochromatic light has the form1

where I (r¯, s)¯ is the radiance (or specific intensity)—average power flux density at point r¯ in the given direction s¯, W/cm2 sr; p(s,¯ s¯ ) is the scattering phase function, 1/sr; and d is the unit solid angle about the direction s¯ , sr. It is assumed that there are no radiation sources inside the medium.

The scalar approximation of the radiative transfer equation (RTE) gives poor accuracy when the size of the scattering particles is much smaller than the wave-

length, but provides acceptable results for particles comparable to and larger than the wavelength.146,184 There is ample literature on the analytical and numerical solutions of the scalar radiative transfer equation.1,3,15,129,130,184–197

If radiative transport is examined in a domain G R3, and ∂G is the domain boundary surface, then the boundary conditions for ∂G can be written in the following general form:

reflection surfaces are present in the domain G, conditions analogous to Eq. (1.10) must be given at each surface.

For practical purposes, integrals of the function I (r,¯ s)¯ over certain phase space regions (r,¯ s)¯ are of greater value than the function itself. Specifically, optical probes of tissues frequently measure the outgoing light distribution function at the medium surface, which is characterized by the radiant flux density or irradiance (W/cm2):

(sN )>

where r¯ ∂G.

In problems of optical radiation dosimetry in tissues, the measured quantity is actually the total radiant-energy-fluence rate U (r)¯ . It is the sum of the radiance over all angles at a point r¯ and is measured by watts per square centimeter:

U (r¯) = I (r,¯ s)¯ d . (1.12)

4π

The phase function p(s,¯ s¯ ) describes the scattering properties of the medium and is in fact the probability density function for scattering in the direction s¯ of a photon traveling in the direction s¯; in other words, it characterizes an elementary scattering act. If scattering is symmetric relative to the direction of the incident wave, then the phase function depends only on the scattering angle θ (angle between directions s¯ and s¯ ), i.e.,

The assumption of random distribution of scatterers in a medium (i.e., the absence of spatial correlation in the tissue structure) leads to normalization:

π

0

In practice, the phase function is usually well approximated with the aid of the postulated Henyey-Greenstein function:1,3,12–16,70,129,130,164

where g is the scattering anisotropy parameter (mean cosine of the scattering angle θ):

π

g ≡ cos θ = p(θ) cos θ · 2π sin θdθ. (1.16)

0

The value of g varies in the range from −1 to 1:145,146 g = 0 corresponds to isotropic (Rayleigh) scattering, g = 1 to total forward scattering (Mie scattering at large particles), and g = −1 to total backward scattering.

The integrodifferential Eq. (1.9) is too complicated to be employed for the analysis of light propagation in scattering media. Therefore, it is frequently simplified by representing the solution in the form of spherical harmonics. Such simplification leads to a system of (N + 1)2 connected differential partial derivative equations known as the PN approximation. This system is reducible to a single dif-

ferential equation of order (N + 1). For example, four connected differential equations reducible to a single diffusion-type equation are necessary for N = 1.191–197

It has the following form for an isotropic medium:

where q(r)¯ is the source function (i.e., the number of photons injected into the unit volume), and

is the reduced (transport) scattering coefficient, 1/cm, and c is the velocity of light in the medium. The transport mean free path of a photon (cm) is defined as

It is worthwhile to note that the transport mean free path (MFP) in a medium with anisotropic single scattering significantly exceeds the MFP in a medium with isotropic single scattering, lt lph [see Eq. (1.8)]. The transport MFP lt is the distance over which the photon loses its initial direction.

Diffusion theory provides a good approximation in the case of a small scat-

tering anisotropy factor g ≤ 0.1 and large albedo → 1. For many tissues, g ≈ 0.6–0.9, and can be as large as 0.990–0.999, for example, for blood.48,49,87,129

This significantly restricts the applicability of the diffusion approximation. It is argued that this approximation can be used at g < 0.9, when the optical thickness τ of an object is of the order 10–20:

d

τ = μtds, (1.23)

0

where d is the tissue depth (thickness) in the direction s.

However, the diffusion approximation is inapplicable for beam input near the object’s surface where single or low-step scattering prevails. When a narrow light beam is normally incident upon a semi-infinite turbid medium with anisotropic scattering, it can be considered as converted into an isotropic point source at the depth of one transport MFP lt [Eq. (1.22)] below the surface. The strength of this point source is the original source strength multiplied by the transport albedo193

It was confirmed that diffusion theory is accurate for describing photon migration in infinite, homogeneous, turbid media.34,46,51,93,191,198–206 However, another

procedure of diffusion equation derivation, described in Refs. 34, 51, and 191, in spite of leading to the basic Eq. (1.20) gives a more general expression for the photon diffusion coefficient:

where a¯ is the numerical coefficient depending on the form of the diffusion equa-

tion (on the scattering anisotropy factor).

Systematic approximation schemes lead to recommendations34,51,191 of a¯ = 0, 1/5, 1/3, 1. Any of these a¯ values gives significantly better agreement with random-walk simulations than the diffusion equation at a¯ = 0, with a¯ = 1/3 being slightly better than the two others.191 Because values a¯ = 1/5 and a¯ = 1 lead to the wrong pulse-front propagation speeds, and only the intermediate value a¯ = 1/3

gives the correct speed, the photon-diffusion coefficient should be taken in the form191

This expression in general gives a better agreement between the diffusion equation and RTE, but in practice it is useful only for highly absorbing tissues or tissue components, when μa/μs > 0.01.199

For accurate use of diffusion theory, one must accurately convert the narrow light beam into isotropic photon sources that must be sufficiently deep in the medium comparable with photon-transport length lt; and the absorption coefficient μa should be much less than the reduced scattering coefficient μs.198

Measurement of diffusely reflected light is often used to infer bulk tissue optical properties for the aims of tissue spectroscopy and imaging. To provide such measurements, an adequate calculating algorithm should be derived. The diffusion

equation solved subject to boundary conditions at the interfaces is one of the bases for the calculation algorithm.46,93,204–206 These boundary conditions are derived

by considering Fresnel’s laws of reflection and balancing the fluence rate and photon current crossing the interface. For the source term modeled as a point scattering source at a depth of one transport MFP, lt, and extrapolated boundary approach sat-

isfying the boundary condition, the spatially resolved steady-state reflectance per incident photon R(rsd) is expressed as205,206

where rsd is the distance between light source and detector at the tissue sur-

face (source-detector separation), cm; r1 = lt2 + rsd2 ; r2 = (lt + 2zb)2 + rsd2 ; zb = 2AD is the distance to the extrapolated boundary; A = (1 + Reff)/(1 − Reff); Reff is the effective reflection coefficient, which can be found by integrating the Fresnel reflection coefficient over all incident angles;204 and D is the diffusion coefficient [see Eq. (1.26)]. The parameters FU and FF represent the fractions of the fluence rate and the flux that exit the tissue across the interface. These values are obtained by integration of the radiance over the backward hemisphere205 and depend on a refractive index mismatch on the boundary.206

Some limitations of the diffusion theory, in particular connected with bad description of the fluence rate if one gets to the source, can be gotten over when it is modified on the basis of accurate but simple Grosjean’s equation, which describes the light distribution in infinite isotropically scattering turbid media.201 A new diffusion approximation to the RTE for a scattering medium with a spatially varying refractive index is derived in Ref. 203.

Now, let us briefly review other solutions of the transport equation. The first-order solution is realized for optically thin and weakly scattering media

(τ < 1, < 0.5), when the intensity of a transmitting (coherent) wave is described by Eq. (1.1) or a similar expression:192

where the incident intensity I0 (W/cm2) is defined by the incident radiant-flux density or irradiance [see Eq. (1.11)] F0 and a solid angle delta function pointed in the direction 0: I0 = F0δ( − 0).

Given a narrow beam (e.g., a laser), this approximation may be applied to denser tissues (τ > 1, < 0.9). However, certain tissues have ≈ 1 in the therapeutic/diagnostic window wavelength range, which makes the first-order approximation inapplicable even at τ 1.

A more strict solution of the transport equation is possible by the discrete ordinates method (multiflux theory) in which Eq. (1.9) is converted into a matrix differential equation for illumination along many discrete directions (angles).183 The solution approximates an exact one as the number of angles increases. It was shown above that the fluence rate can be expanded in powers of spherical harmonics, separating the transport equation into components for spherical harmonics. This approach also leads to an exact solution, provided the number of spherical harmonics is sufficiently large. For example, a study of tissues made use of up to 150 spherical harmonics,207 and the resulting equations were solved by the finite-difference method.208 However, this approach requires tiresome calculations if a sufficiently exact solution is to be obtained. Moreover, it is hardly suitable for δ-shaped phase scattering functions.212

The P3-approximation is an approximate solution to the RTE [see Eq. (1.9)], which expresses the radiance algebraically in a truncated series of Legendre poly-

nomials. Star was the first to use the advances in computer power to compare the P3-approximation to Monte Carlo calculations in a slab geometry.209,210 The fur-

ther development of the P3-approximation for a spherical geometry that is more practical for application in tissue study is described in Ref. 211.

Tissue optics extensively employs simpler methods for the solution of transport

equations, e.g., the two-flux Kubelka-Munk theory212 or three-, four-, and sevenflux models.56,183,192 Such representations are natural and very fruitful for laser tis-

sue probing. Specifically, the four-flux model213 is actually two diffuse fluxes traveling to meet each other (Kubelka-Munk model) and two collimated laser beams, the incident one and the one reflected from the rear boundary of the sample. The seven-flux model is the simplest three-dimensional representation of scattered radiation and an incident laser beam in a semi-infinite medium.56 Of course, the simplicity and the possibility of expeditious calculation of the radiation dose or rapid determination of tissue optical parameters (solution of the inverse scattering problem) is achieved at the expense of accuracy.

1.1.3 Monte Carlo simulation techniques

The development of new methods for the solving forward and inverse radiation transfer problems in media with arbitrary configurations and boundary conditions is crucial for the reliable layer-by-layer measurements of laser radiation inside tissues and is necessary for practical purposes such as diffuse optical tomography and the spectroscopy of biological objects. The Monte Carlo (MC) method appears to be especially promising in this context, being widely used for the numerical solution of the RTT equation in different fields of knowledge (astro-

physics, atmosphere and ocean optics, etc.).214 It has recently been applied to tissue optics.1–3,12–16,29,33,41,198,213,215–247 The method is based on the numerical simula-

tion of photon transport in scattering media. Random migrations of photons inside a sample can be traced from their input until absorption or output. Known algorithms allow a few tissue layers with different optical properties to be characterized along with the final incident beam size and the reflection of light at interfaces. Typical examples of multilayer tissues are skin, vascular tissue, urinary bladder, and uterine walls.

For all its high accuracy and universal applicability, the MC method has one major drawback, which is that it consumes too much computation time needed to trace a large number of photons to get an acceptable variance due to the statistical nature of modeling. The MC simulations are especially computationally expensive when the absorption coefficient is much less than the scattering coefficient of the media, in which photons may propagate over a long distance before being absorbed.

Depending on the problem to be solved, the MC technique is used to either simulate the diffuse reflectance or transmittance for one wavelength or for a whole

spectrum; other optical characteristics at various experimental geometries also can be modeled.1–3,12–16,29,33,41,198,213,215–247 Because the implicit photon capturing

technique is used during the MC simulation, a photon packet with an initial weight of unity is launched perpendicularly to the tissue surface along the direction of the light beam for the problem of pencil beam propagation, and isotropically for the problem of light distribution of an isotropic light source inserted into a tissue.

Other geometries are also possible. Then, a step size is chosen statistically using the expression198

where ξ is a random number equidistributed between 0 and 1 (0 < ξ ≤ 1). Because of absorption in the system, the photon packet loses some of its weight at the end of each step. The amount of weight lost is the photon weight at the beginning of the step multiplied by (1 − ), where is the albedo [see Eq. (1.7)]. The photon with the remaining weight is scattered. A new photon direction is statistically determined by a phase function [see Eq. (1.13)], which according to the scattering anisotropy factor g can be taken in the form of the Henyey-Greenstein postulated

function [see Eq. (1.15)]. A new step size is then generated by Eq. (1.29), and the

process is repeated. When the photon does try to leave the medium, the probability of an internal reflection is calculated using Fresnel’s equation.204,230 When the

photon weight is less than a preset threshold (usually 10−4), a form of “Russian roulette” is used to determine whether the photon should be terminated or propagated further with an increased weight. If the photon packet crosses the surface boundary into the ambient medium, the photon weight contributes to the diffuse reflectance or transmittance. If a reflection occurs, then the photon packet is reflected back into the medium the appropriate distance and migration continues. Otherwise, the migration of that particular photon packet halts and a new photon is launched into the medium at the predefined source location. Multiple photon packets are used to obtain statistically meaningful results; at present 1–10 million photon packets are usually used. For example, three-dimensional MC code, designed for photon migration through complex heterogeneous media, allows one to obtain a SNR greater than 100 up to distances of 30 mm with a 1 mm2 detector

with 108 photons propagated within 5–10 hr of computer time on a Pentium III 1000 MHz CPU.245

Although advanced computer facilities and software systems have reduced the time needed, further developments in laser diagnostic and therapeutic tools require more effective, relatively simple, and reliable algorithms of the MC method. For instance, the condensed MC method allows one to obtain the solution for any albedo based on the results of modeling for a single albedo, which substantially facilitates computation.226 Also, the development of very economical hybrid models currently underway is intended to combine the accuracy of the MC

method and the high performance of diffusion theories or approximating analytic

expressions.198,225,229,230

The original MC code that allows one to obtain information required to reconstruct an internal structure of highly scattering objects with size of 1000 scattering lengths and more was recently designed based on the path-integration technique and Metropolis algorithm.248 The path-integral apparatus first suggested by Feynman for the alternative description of quantum mechanics can be also used to describe the movement of photons in a turbid medium as if they are particles under-

going collisions at a given collision frequency with a mean deflection in trajectory per collision.249–254 The integral over all possible paths using a set of nested

integrals is called a “path integral.” This approach offers analytical solutions to the RTE in the framework of Perelman’s approximation that is valid for a relatively weak scattering.250 The path-integral model described in Ref. 249 is derived from first principles and does not include Perelman’s approximation. The pathintegral technique was applied to numerical calculations in the model of photon “random walk” within a three-dimensional discrete grid.254 In the context of the MC approach, the path-integral technique may be seen as an extreme form of vari-

ance reduction, when instead of finding the most likely paths by random sampling, the path-integral formalism sets out to identify them directly.248,249 Therefore, the

elimination of “uninformative” photon paths from the calculations may provide a few-orders-higher calculation rate.248

Let us consider human-skin optics as an example.37,38,57,213,221,222,224,227,228, 236,237,243,246,255–262 In order to calculate distributions of the radiant-flux den-

sity F (r¯) and the total radiant-energy-fluence rate U (r)¯ by the MC method [see Eqs. (1.11) and (1.12)], let us represent the skin as a plane multilayer scattering and absorbing medium (Fig. 1.8), with a laser beam falling normally onto its surface. Let us further assume that each ith layer is characterized by the following parameters: μai , μsi , pi (θ), the thickness di , and the refractive index of the filler medium ni . It should be noted that a more general approach to MC simulation that

accounts for the interfaces between the dermal layers as quasi-random periodic surfaces and spectral skin response is also available.243,246

Figure 1.8 A model of human skin.213

Using the MC algorithm described in Refs. 213, 224, 261, and 262 to simulate the distribution of Gaussian light beams in the skin (see Fig. 1.8 and Table 1.1), the total fluence rate at wavelengths 337, 577, and 633 nm was obtained as shown in Fig. 1.9, along with the dependencies of the maximum total radiant-energy-fluence rate U m and the maximum fluence rate area D × D on the incident beam radius r0 at 633 nm (Fig. 1.10). D and D are defined at the 1/e2 level of U along and across the incident light beam, respectively. It is readily seen that the illumination maximum is formed at a certain depth inside the tissue, and the total fluence rate at the point of maximum Um may be significantly higher than that in the middle of the beam incident to the surface of the medium (U0). This was noticed by many authors (see, for instance, Refs. 1, 3, 37, and 210), who emphasized the strong correlation between the Um/U0 ratio and the optical properties of the medium, the incident beam radius, and boundary properties. It appears from Fig. 1.10(b) that an

Figure 1.9 Results of Monte Carlo simulation of the total radiant-energy-fluence rate distribution U (W/cm2) in skin irradiated by Gaussian laser beams with different wavelengths (λ = 633, 577, and 337 nm), equal radius on the skin surface (r0 = 1.0 mm), and equal intensity at the beam center (U0 = 1 W/cm2).6 z is the linear coordinate (depth inside the skin) and r is the coordinate across the light beam.

Table 1.1 Optical parameters of skin.6,213

increase in the incident beam radius leads to a broadening of the illuminated area inside the tissue, with the enhancement rate in the transversal direction exceeding that along the beam.

For practical purposes, such calculations for human skin and other multilayered soft tissues are necessary to correctly choose the irradiation doses for photochemical, photodynamic, and photothermal therapy of cancer and many other

(a)

(b)

Figure 1.10 Parameters of the maximum illumination area as functions of the incident beam radius. A beam with a Gaussian profile, wavelength 633 nm, power 25 mW. (a) 1, Total illumination in the center of the incident beam U0; 2, maximal total illumination Um; 3, Um/U0.

(b) 1, The size of the maximally illuminated area (at the 1/e2 level) along the beam axis, D ; 2, size of the maximally illuminated area (at the 1/e2 level) across the beam axis D .213

diseases, and the laser coagulation of the superficial blood vessels or transscleral

cyclophotocoagulation.2,3,10–16,22,29,32,33,37,57,72,90,91,210,258–268

In particular, based on the results of MC simulation presented in Figs. 1.9 and 1.10, attenuation of a wide laser beam of intensity I0 at depths z > ld = 1/μeff [see Eq. (1.18)] in a thick tissue may be described as

where bs accounts for additional irradiation of upper layers of a tissue due to backscattering (photon recycling effect). Respectively, the depth of light penetra-

Typically, for tissues bs = 1–5 for beam diameter of 1–20 mm.210,265 Thus, when wide laser beams are used for irradiation of highly scattering tissues with lowabsorption, CW light energy is accumulated in tissue due to the high multiplicity of chaotic long-path photon migrations. A highly scattering medium works as a random cavity, providing the capacity of light energy. The light power density within the superficial tissue layers may substantially (up to fivefold) exceed the incident power density and cause the overdosage during photodynamic therapy or overheating during interstitial laser thermotherapy. On the other hand, the photon recycling effect can be used for more effective irradiation of undersurface lesions at relatively small incident power densities.

1.2 Short pulse propagation in tissues

1.2.1 Basic principles and theoretical background

Based on the time-dependent radiation-transfer theory (RTT), it is possible to analyze the time response of scattering tissue. Such an analysis is important

tissues.1,3,31,42,44,71,92,129,130,199,200,203,205,248–254,269–302 In its general form, the

time-dependent RTT equation for time-dependent radiance (or the specific intensity) I (r¯, s,¯ t) can be written as:269,301

Compared with the CW equation (1.9), the following notation is introduced into Eq. (1.32): t is time, t2 = l/(μtc) is the average interval between interactions, where c is the velocity of light in the medium; f (t, t ) describes the temporal deformation of a δ-shaped pulse following its single scattering and can be represented in the form of an exponentially decaying function as

where t1 may be a function of r¯, and t1 is the first moment of the distribution function f (t, t ) that describes the time interval of an individual scattering act at t1 → 0, f (t, t ) → δ(t − t ). The radiance I (r,¯ s,¯ t) in Eq. (1.32) contains two

components: the attenuated incident radiation and the diffuse. This equation meets the boundary conditions [see Eq. (1.10)] at (r¯, s)¯ → (r¯, s,¯ t).

When probing the plane-parallel layer of a scattering medium with an ultrashort laser pulse, the transmitted pulse consists of a ballistic (coherent) component, a group of photons having zigzag trajectories, and a highly intensive diffuse component (see Fig. 1.11).1,3,31 Both unscattered photons and photons undergoing forward-directed single-step scattering contribute to the intensity of the ballistic component (composed of photons traveling straight along the laser beam). This component is subject to exponential attenuation with increasing sample thickness [see Eq. (1.1)]. This accounts for the limited utility of ballistic photons for practical diagnostic purposes in medicine.

Figure 1.11 An ultrashort laser pulse propagating through a random medium spreads into a diffuse component, a snake with zigzag paths, and a ballistic component.1

The group of snake photons with zigzag trajectories includes photons that have experienced only a few collisions each. They propagate along trajectories that deviate only slightly from the direction of the incident beam and form the first-arriving part of the diffuse component. These photons carry information about the optical properties of the random medium and parameters of any foreign object that they may happen to come across during their progress.

The diffuse component is very broad and intense since it contains the bulk of incident photons after they have participated in many scattering acts and therefore migrate in different directions and have different path lengths. Moreover, the diffuse component carries information about the optical properties of the scattering medium, and its deformation may reflect the presence of local inhomogeneities in the medium. The resolution obtained by this method at a high light-gathering power is much lower than in the method measuring straight-passing photons. Two probing schemes are conceivable, one recording transmitted photons and the other taking advantage of their backscattering (see Fig. 1.12).

If in the diffusion approximation (valid at μa μs) the tissue is homogeneous and semi-infinite, the size of both the source and the detector is small compared with the distance rsd between them at the tissue surface, and the pulse may be

Figure 1.12 Typical schemes for time-resolved tissue studies:1 (a) Recording transmitted photons; (b) backscattering regime. A, Probing beam; B, detected radiation. The dark area in the center of the scattering layer is a local inhomogeneity (tumor). Spatial and temporal photon distributions in the medium are shown.

regarded as single; then the light distribution is described by the time-dependent diffusion equation:272,273

which is in fact the generalization of the CW Eq. (1.17). It is worth noting that the diffusion equation is equivalent to the equation for thermal conductivity.286 The

solution of Eq. (1.34) yields the following relation for the number of backscattered photons at the surface for unit time and from unit area R(rsd, t):272,273

the shape of a pulse measured by the time-resolved photon counting technique. Experimentally measured optical parameters of many tissues and model media obtained by the pulse method can be found in Refs. 1, 3, 6, 12–15, 31, 88, 89, 129,

130, 245, 272–289, and 300. An important advantage of the pulse method is its applicability to in vivo studies owing to the possibility of the separate evaluation of μa and μs using a single measurement in the backscattering or transillumination regimes. It seems appropriate to mention that a search for more adequate approaches to describing tissue responses to laser pulses is underway (see, for instance, Refs. 199, 200, 203, 204, 248–254, and 291–302). Many publications are devoted to image transfer in tissues and the evaluation of the resolving power of

optical tomographic schemes that make use of the first-transmitted photons of ul-

trashort pulses.1,3,71,129,130,245,270,271,279–284,294,297–302

1.2.2Principles and instruments for time-resolved spectroscopy and imaging

The main principle of enhanced viewing through a turbid medium (tissue) using a time-resolved approach is well illustrated in Fig. 1.13.1,31 A contrast image of an object in a scattering medium can be provided by electronic or optical time-

gating of the earliest-arriving, minimally scattered light (ballistic and snake photons), which contains geometric information.1,3,31,71 The typical optical schemes

using the selection of the earliest-arriving photons are presented in Figs. 1.14 and 1.15. The first group of schemes (see Fig. 1.14) uses the electronic time-gating procedure. The time-correlated single-photon-counting technique explores a high- repetition-rate picosecond laser (for example, a cavity-dumped mode-locked dye laser). At the detection of the earliest-arriving photons, the time delay is measured with a time-to-amplitude converter [see Fig. 1.14(a)] and a histogram of the arrival

Figure 1.13 Gated viewing through tissue. An enhanced spatial resolution is obtained by selecting “early” light only.1

(a)

(b)

Figure 1.14 Techniques for electronically gated viewing.1

times is built up using a large number of low-energy pulses. The time resolution of such a technique is limited to about 50 ps. For more energetic pulses from lasers with a lower repetition rate, the use of streak cameras allows for a time resolution down to 1 ps [see Fig. 1.14(b)]. If a synchroscan streak camera is employed, even a high-repetition rate source with low-energy pulses can be used.

The second group of techniques uses optical nonlinear effects to select photons (see Fig. 1.15).1 For a scheme with an optical Kerr gate, an energetic laser is used. Part of the pulse is transmitted into the tissue and part opens the shutter by use of the optical Kerr effect (the cell with CS2) [see Fig. 1.15(a)]. Since the gate width is determined only by the length of the laser pulse, subpicosecond gate times can be achieved. The Raman-amplifier-gating technique also uses energetic laser pulses. A Stokes wave generated by stimulated Raman scattering in a gas cell is used to probe the tissue [see Fig. 1.15(b)]. The low-intensity transmitted light is amplified in a Raman amplifier, which in turn is pumped by an ultrashort laser pulse. This pulse has the proper time delay to strobe on the desired early temporal part of the light under investigation. The third scheme uses a time-correlated frequencydoubling technique and is often used in optical autocorrelators for monitoring laser pulse characteristics [see Fig. 1.15(c)]. It can be used directly for optical gating of signal photons.

1.2.3 Coherent backscattering

The use of ultrafast laser pulses gives rise to a local peak of intensity backscattered within a narrow solid angle owing to scattered light interference.31,73,74 In

the exact backward direction, the intensity of the scattered light is normally twice

(a)

(b)

(c)

Figure 1.15 Gated-viewing technique using nonlinear optical phenomena:1 CS2 is the optical Kerr cell filled by CS2; KDP is the nonlinear crystal for frequency doubling.

the diffuse intensity. Such interference in coherence arises from the time reversal symmetry among various scattered light paths in the backscattering direction. This phenomenon is known as weak localization. The profile of the angular distribution of the coherent peak depends on the transport mean path lt and the absorption coefficient μa. The angular width of the peak is directly related to lt as74

In many hard and soft tissues such as human fat tissue, lung cancer tissue, normal and cataractous eye lens, and myocardial, mammary, and dental tissues, the

backscattered coherent peak occurs when the probing laser pulse is shorter than 20 ps.74

1.3 Diffuse photon-density waves

1.3.1 Basic principles and theoretical background

The frequency-domain (FD) method has been proposed for photon migration studies in scattering media. The method is designed to evaluate the dynamic response of scattered light intensity to modulation of the incident laser beam

intensity in a wide frequency range, usually in tissue research using 50 to

1000 MHz.1,3,4,6,10,52,53,71,129,130,285,286,301–337 The FD method measures the mod-

ulation depth of scattered light intensity mU ≡ acdetector/dcdetector (see Fig. 1.16) and the corresponding phase shift relative to the incident light modulation

Figure 1.16 Schematic representation of the time evolution of the light intensity measured in response to (a) a very narrow light pulse and (b) a sinusoidally intensity-modulated light transversing an arbitrary distance in a scattering and absorbing medium. (a) If the medium is strongly scattering, there are no unscattered components in the transmitted pulse. (b) The transmitted photon density wave retains the same frequency as the incoming wave in the medium. The reduced amplitude of the transmitted wave arises from attenuation related to the scattering and absorption processes. The demodulation is the ratio ac/dc normalized to the modulation of the source.314

phase (phase lag). Compared with the time-domain (TD) measurements described earlier, this method is more simple and reliable in terms of data interpretation and immunity from noise. These happen because FD equipment involves amplitude modulation at low peak power, slow rise time [compare Figs. 1.16(a) and 1.16(b)], and hence smaller bandwidths than TD instruments; higher SNRs are attainable as well. Medical FD equipment is more economic and portable, and can be built on the basis of measuring devices used in optical telecommunication systems and studies of optical fiber dispersion.4,328 However, the FD technique suffers from the simultaneous transmission and reception of signals and requires special attempts to avoid unwanted cross talk between the transmitted and detected

signals. The current measuring schemes are based on heterodyning of optical and

transformed signals.1,3,4,6,10,52,53,71,129,130,285–290,301–323,334,335

The development of the theory underlying this method resulted in the discovery of a new type of wave: photon-density waves or waves of progressively decaying intensity. Microscopically, individual photons make random migrations in a scattering medium, but collectively they form a photon-density wave at a modulation frequency ω that moves away from a radiation source (see Figs. 1.16 and 1.17). Diffuse waves of this type are well known in other fields of physics (for example,

thermal waves are excited upon absorption of modulated laser radiation in various media, including biological ones5,6,25). Photon-density waves possess typical wave

properties; e.g., they undergo refraction, diffraction, interference, dispersion, and

attenuation.1,3,4,6,71,52,53,129,130,285,301,303,307–310,313–315,334

Figure 1.17 Schematic representation of photons propagation in scattering media induced by (a) a CW, (b) a pulse, and (c) a sinusoidally intensity-modulated light source.315

In strongly scattering media with weak absorption far from the walls and a source or a receiver of radiation, the light distribution may be regarded as a decaying diffusion process described by the time-dependent diffusion equation for photon density [see Eq. (1.34)]. When a point light source with harmonic intensity modulation is used, placed at the point r¯ = 0,

where mI is the intensity modulation depth of the incident light.

The solution of Eq. (1.34) for a homogeneous infinite medium can be presented in the form71

Eq. (1.18)], and kr(ω) and ki(ω) are the real and imaginary parts of the photondensity wave vector, respectively:

where τa = 1/(μac) is the average travel time of a photon before being absorbed. An alternating component of this solution is a retreating spherical wave with its center at the point r¯ = 0 that oscillates at a modulation frequency ν and undergoes

a phase shift relative to the phase value at point r¯ = 0 equal to

Constant and time-dependent components of the photon-density wave fall with distance as exp(−r/¯ ld) and exp[−ki(ω)r¯], respectively. The length of a photondensity wave is defined by

It clearly follows from Eq. (1.50) that in order to support the transport of a photondensity wave in a medium, light scattering is needed (see first exponential term); but in contrast, high scattering turns a photon density wave to decay (see second exponential term).

Measuring mU (r¯, ω), (r,¯ ω) allows one to separately determine the transport scattering coefficient μs and the absorption coefficient μa, and evaluate the spatial distribution of these parameters.

1.3.2Principles of frequency-domain spectroscopy and imaging of tissues

Evidently, there is a close relationship between the two time-resolved methods for the assessment of optical properties of tissues. In the case of pulse probing of a scattering medium, Fourier instrumental or computer-aided analysis of scattered pulses

allows us to simultaneously obtain the amplitude-phase response of the medium for a continuous set of harmonics.1,3,4,71,286,301–303,311 Figure 1.18 illustrates the typ-

ical behavior of the amplitude-phase response in a tissuelike phantom (whole or

diluted milk).71 Such characteristics are useful for the spectroscopic examination of tissues, e.g., for in vivo evaluation of hemoglobin oxygenation321,338 or blood glucose level.339–341

The spatial resolution available using photon-density waves is crucial for the visualization of macroinhomogeneities. Theoretical considerations illustrated by

Figure 1.18 (a) Amplitude and (b) phase responses of a model medium [unskimmed (1, 3) and 40% diluted (2) milk] obtained by the Fourier transformation of experimental pulse responses; 1, 2, recording transmitted pulses, 2-cm-thick cuvette; 3, backscattering regime (large volume of unskimmed milk). The distance between irradiating and detecting optic fibers is rsd = 2 cm.71

Fig. 1.19 for two absorbing macroinhomogeneities in a scattering medium provide

evidence that their separate identification is feasible if the accuracy of the phase and wave amplitude measurements is not less than 1.0 and 2.0%, respectively.310,330

The predicted resolving power of diffuse tomography using photon-density waves is close to 1 mm; i.e., it is comparable to that of positron-emission and magnetoresonance tomography.4,285 Important advantages of optical tomography are technical simplicity, the enhancement of an object’s contrast by molecular dyes, and the visualization of local metabolic processes.

Figure 1.20 presents images of tumor-containing female breast tissue outlined by contour lines for μa and μs that were obtained by exposure to modulated visible and near-IR radiation.71 The tumor is readily discernible because of its high absorption and scattering coefficients.

In principle, a record-breaking resolving power of less than 1 mm can be

achieved by taking advantage of the interference of photon-density waves excited by spaced sources.4,53,302,330,342 Not only a good spatial resolution, but also a high

contrast and low sensitivity to the movements and geometry of the object under

study should be provided to get a high-quality image. A tissue immersion technique can be used to improve image quality.129,324,328 When the imaged tissue

is surrounded with a medium of matched scattering and absorption properties, a uniform transillumination image is obtained and the maximum dynamic range is available to examine variations within the tissue. In addition, it allows one to eliminate the influence of the boundary conditions and geometry for objects with a

Figure 1.19 Theoretical distributions of the relative phase shift and intensity modulation at 200 MHz in an irradiated system of two absolute absorbers (balls 0.5 cm in diameter) placed in a homogeneous scattering medium (μs = 10 cm−1, μa = 0.02 cm−1). The source is located at the origin and the absorbers are located at points (−2, 4) and (2, 4).310

Figure 1.20 Reconstructed optical images of human cancerous breast tissue obtained by exposure to modulated radiation in the visible and NIR wavelength regions. (a) The image outlined by contour lines for the absorption coefficient μa (10−2 cm−1). (b) The same for the transport scattering coefficient μs (cm−1). The tumor is located near the point (70, 10); the coefficients have relative values.71

complex shape and structure. It also allows one to achieve optical matching of the probe laser beam and tissue, to considerably reduce the influence of external and internal movements (breathing, heartbeats, muscle tremor, etc.) of the object during the imaging process, and to calibrate measurements using well-known optical properties of the immersion medium. Sometimes, such a simple technique is a good alternative for more sophisticated imaging techniques based on multifrequency analysis.

Apart from the visualization of macroinhomogeneities in breast tissues, the FD method is useful in examining other tissue, e.g., brain and lungs. It also provides an insight into many physiological processes dependent on oxygen consumption by tissues and organs and related hemodynamic changes. One of the most important

areas of application may be the evaluation of the oxygen distribution in a functioning brain.4,285,302,303 Another example is the monitoring of neoplastic growth

patterns, including enhanced blood volume and blood deoxygenation, increased in-

tracellular organelle content, and tissue calcification, which may be important for the differentiation between benign and malignant tumors.285,302,303

To conclude, it should be emphasized that high tissue density sometimes necessitates taking into consideration the time interval of an individual scattering act t1

[see Eqs. (1.32) and (1.33)], which may prove comparable to the mean time interval between interactions t2.301,327 Moreover, the widely used diffusion approximation

imposes important constraints on the analysis of the optical properties of tissues.

Therefore, the development of more universal models of photon-density wave dispersion is well under way using new MC algorithms.301–303,327

1.4 Propagation of polarized light in tissues

1.4.1 Introduction

Up to this point, we have ignored the vector nature of light transport in scattering media such as tissues because we assumed it to be rapidly depolarized during propagation in a randomly inhomogeneous medium. It is a common belief that the randomness of tissue structure results in fast depolarization of light propagating in tissues. Therefore, polarization effects are usually ignored. However, in certain tissues (transparent tissues such as eye tissues, cellular monolayers, mucous membrane, and superficial skin layers), the degree of polarization of transmitted or reflected light remains measurable, even when the tissue has a considerable thickness. In such a situation, the information about the structure of tissues and cell ensembles can be extracted from the registered depolarization degree of initially polarized light, the polarization

state transformation, or the appearance of a polarized component in the scattered

light.3,5,6,59,67–70,105,129,135,138,145,146,148,149,155,159,166,186,343

In regard to practical implications, polarization techniques are believed to give rise to simplified schemes of optical-medical tomography compared with time-

resolved methods, and also provide additional information about the structure of tissues.343–396

1.4.2 Tissue structure and anisotropy

Many biological tissues are optically anisotropic.3,9,10,24,29,43,59–70,97,127–130,135, 138,150,151,166,168,176,177,397–440 Tissue birefringence results primarily from the lin-

ear anisotropy of fibrous structures, which forms the extracellular media. The refractive index of a medium is higher along the length of a fiber than along the cross section. A specific tissue structure is a system composed of parallel cylinders that create a uniaxial birefringent medium with the optic axis parallel to the cylinder axes. This is called birefringence of form. A large variety of tissues such as eye cornea, tendon, cartilage, eye sclera, dura mater, testis, muscle, nerve, retina, bone, teeth, myelin, etc. exhibit form birefringence. All of these tissues contain uniaxial and/or biaxial birefringent structures. For instance, in bone and teeth, these are mineralized structures originating from hydroxyapatite crystals, which play an important role in hard tissue birefringence. In particular, dental

enamel is an ordered array of such crystals surrounded by a protein/lipid/water matrix.65,66,97,423,425,426 Fairly well oriented hexagonal crystals of hydroxyapatite

of approximately 30–40 nm in diameter and up to 10 μm in length are packed into an organic matrix to form enamel prisms (or rods) with an overall cross section of 4–6 μm. Enamel prisms are roughly perpendicular to the tooth surface. Tooth dentin is a complex structure, honeycombed with dentinal tubules, which

are shelled organic cylinders with a highly mineralized shell. Tubules diameters are 1–5 μm, and their number density is in the range (3–7) × 106 cm−2.65,66,97,423

Tendon consists mostly of parallel, densely packed collagen fibers arranged

in parallel bundles interspersed with long, elliptical fibroblasts. In general, tendon fibers are cylindrical in shape with diameters ranging from 20 to 400 nm.176,177

The ordered structure of collagen fibers running parallel to a single axis makes tendon a highly birefringent tissue.

Arteries have a more complex structure than tendons. The medial layer consists mostly of closely packed smooth muscle cells with a mean diameter of 15–20 μm. Small amounts of connective tissue, including elastin, collagenous, and reticular fibers, as well as a few fibroblasts, are also located in the media. The outer adventitial layer consists of dense fibrous connective tissue. The adventitia is largely made up of collagen fibers, 1–12 μm in diameter, and thinner elastin fibers, 2–3 μm in diameter. As with tendon, the cylindrical collagen and elastin fibers are ordered mainly along one axis, thus causing the tissue to be birefringent.

Myocardium, on the other hand, contains fibers oriented along two different axes. Myocardium consists mostly of cardiac muscle fibers arranged in sheets that wind around the ventricles and atria. In pigs, the myocardium cardiac muscle is comprised of myofibrils (about 1 μm in diameter) that in turn consist of cylindrical myofilaments (6–15 nm in diameter) and aspherical mitochondria (1–2 μm in di-

ameter). Myocardium is typically birefringent since the refractive index along the axis of the muscle fiber is different from that in the transverse direction.176,177

Form birefringence arises when the relative optical phase between the orthogonal polarization components is nonzero for forward-scattered light. After multiple

forward scattering events, a relative phase difference accumulates and a delay (δoe) similar to that observed in birefringent crystalline materials is introduced between orthogonal polarization components. For organized linear structures, an increase in phase delay may be characterized by a difference ( noe) in the effective refractive index for light polarized along, and perpendicular to, the long axis of the linear structures. The effect of tissue birefringence on the propagation of linearly polarized light is dependent on the angle between the incident polarization orientation and the tissue axis. Phase retardation δoe between orthogonal polarization components is proportional to the distance d traveled through the birefringent medium:416

A medium of parallel cylinders is a positive uniaxial birefringent medium [ noe = (ne − no) > 0] with its optic axis parallel to the cylinder axes [see Fig. 1.21(a)]. Therefore, a case defined by an incident electrical field directed parallel to the cylinder axes will be called “extraordinary,” and a case with the incident electrical field perpendicular to the cylinder axes will be called “ordinary.” The difference (ne − no) between the extraordinary index and the ordinary index is a measure

of the birefringence of a medium comprised of cylinders. For the Rayleigh limit (λ cylinder diameter), the form birefringence becomes399,402

where f1 is the volume fraction of the cylinders; f2 is the volume fraction of the ground substance; and n1, n2 are the corresponding indices. For a given index difference, maximal birefringence is expected for approximately equal volume fractions of thin cylinders and ground material. For systems with large diameter cylinders (λ cylinder diameter), the birefringence goes to zero.402

Figure 1.21 Models of tissue birefringence: (a) system of long dielectric cylinders, (b) system of thin dielectric plates.442

For a system of thin plates [see Fig. 1.21(b)], the following equation is

where f1 is the volume fraction occupied by the plates; f2 is the volume fraction of the ground substance; and n1, n2 are the corresponding indices. This implies that the system behaves like a negative uniaxial crystal with its optical axis aligned normally with the plate surface.

Form birefringence is used in biological microscopy as an instrument for studying cell structure. The sign of the observed refractive index difference points to the particle shape closest to that of the rod or the plate, and if n1 and n2 are known, one can then assess the volume fraction occupied by the particles. To separate the birefringence of the form and the particle material, the refractive indices of the particles and the ground substance should be matched, because form birefringence vanishes with n1 = n2.

Linear dichroism (diattenuation), i.e., different wave attenuation for two orthogonal polarizations, in systems formed by long cylinders or plates is defined by the difference between the imaginary parts of the effective indices of refraction. Depending on the relationship between the sizes and the optical constants of the cylinders or plates, this difference can take both positive and negative values.160

Reported birefringence values for tendon, muscle, coronary artery, myocardium, sclera, cartilage, and skin are on the order of 10−3 (see, for instance, Refs. 400, 409, 410, and 412–416). The measured refractive index variations for the fast and slow axes of rabbit cornea show that its birefringence varies within the

range of 0 at the apex, or top of the cornea, to 5.5 × 10−4 at the base of the cornea, where it attaches to the sclera.398,404 The predominant orientation of collagenous

fibers in different regions of the cornea results in birefringence and dichroism.403 Based on experimental results, it has been assumed that the birefringent portions of the corneal surface all have a relatively universal fast axis located approximately 160 deg from the vertical axis, defined as a line that runs from the apex of the cornea through the pupil.404

A new technique—polarization-sensitive optical-coherence tomography (PS

OCT)—allows for the measurement of linear birefringence in turbid tissue with high precision.412–416,418 The following data have been reported using this tech-

nique: for rodent muscle, 1.4 × 10−3 (Refs. 415 and 416); for normal porcine tendon, (4.2 ± 0.3) × 10−3 and for thermally treated (90◦C, 20 s), (2.24 ± 0.07) × 10−3; for porcine skin, 1.5 × 10−3–3.5 × 10−3; for bovine cartilage, 3.0 × 10−3 (Ref. 418); and for bovine tendon, (3.7 ± 0.4) × 10−3 (Ref. 413). Such birefringence provides 90% phase retardation at a depth on the order of several hundred micrometers.

The magnitude of birefringence and diattenuation are related to the density and other properties of the collagen fibers, whereas the orientation of the fast axis indicates the orientation of the collagen fibers. The amplitude and orientation of

birefringence of the skin and cartilage are not as uniformly distributed as in tendon. In other words, the densities of collagen fibers in skin and cartilage are not as uniform as in tendon, and the orientation of the collagen fibers is not distributed in as orderly a fashion.418

In addition to linear birefringence and dichroism (diattenuation), many tissue components show optical activity. In polarized light research, the molecule’s chi-

rality, which stems from its asymmetric molecular structure, results in a number of characteristic effects generically called optical activity.430,434 A well-known man-

ifestation of optical activity is the ability to rotate the plane of linearly polarized light about the axis of propagation. The amount of rotation depends on the chiral molecular concentration, the path length through the medium, and the light wavelength. For instance, chiral asymmetrically encoded in the polarization properties of light transmitted through a transparent media enables very sensitive and accu-

rate determination of glucose concentration. Tissues containing chiral components display optical activity.404,428,429 Interest in chiral turbid media is driven by the

attractive possibility of noninvasive in situ optical monitoring of the glucose in diabetic patients.105,138 Within turbid tissues, however, where the scattering effects

dominate, the loss of polarization information is significant and the chiral effects due to the small amount of dissolved glucose are difficult to detect.

In complex tissue structures, chiral aggregates of particles, in particular spherical particles, may be responsible for optical activity of tissue (see Fig. 1.22). More sophisticated anisotropic tissue models can also be constructed. For example, the cornea can be represented as a system of plane anisotropic layers (plates, i.e., lamellas), each of which is composed of densely packed long cylinders (fibrils) [see Fig. 1.21(a)] with their optical axes oriented along a spiral (see Fig. 2.2). This fibrilar-lamellar structure of the cornea is responsible for the linear and circular dichroism and its dependence on the angle between the lamellas.403

Figure 1.22 Examples of chiral aggregates of spherical particles.

1.4.3 Light scattering by a particle

Let us now consider the transformation of any polarization type (linear, circular, or elliptical) in a scattering medium with typical tissue parameters, and compare the penetration depth of circular and linear polarization in different media. To this end, let us examine a monochromatic plane wave incident on an isolated

respectively, to the scattering plane. The scattered electrical-field vector is given by

There is a linear relationship between the incident and scattered field components, defined by Eqs. (1.55) and (1.57):43,148,149

where k = 2π/λ is the wave number, λ = λ0/n¯ is the wavelength in the scatter-

ing medium; n¯ is the mean refractive index of the scattering medium; λ is the

√ 0 wavelength of the light in vacuum; i = −1; r is the distance from the scatterer

to the detector; and z is the position coordinate of the scatterer. The complex num-

bers S1–4 are the elements of the amplitude scattering matrix (S-matrix) or Jones matrix.43,148,149,160,443–449 They each depend on scattering and azimuthal angles θ

and ϕ, and contain information about the scatterer. Both amplitude and phase must be measured to quantify the amplitude scattering matrix. The direct measurements of matrix elements can be done using a two-frequency Zeeman laser, which produces two laser lines with a small frequency separation (about 250 kHz) and orthogonal linear polarizations,149 or by the coherence optical tomography (OCT) technique.418

1.4.4 Polarized light description and detection

Definitions of polarized light and its properties, as well as production and detection techniques, are well described in literature.135,160,443–448 Polarization refers to the

pattern described by the electric-field vector as a function of time at a fixed point in space. When the electrical-field vector oscillates in a single, fixed plane all along the beam, the light is said to be linearly polarized. This linearly polarized wave can be resolved into components parallel and perpendicular to the scattering plane [see Eqs (1.55) and (1.57) and Fig. 1.23]. If the plane of the electrical field rotates, the light is said to be elliptically polarized because the electrical-field vector traces out an ellipse at a fixed point in space as a function of time. If the ellipse happens to be a circle, the light is said to be circularly polarized. The connection between phase and polarization can be understood as follows: circularly polarized light consists of equal amounts of linear mutually orthogonal polarized components that oscillate exactly 90 deg out of phase. In general, light of arbitrary elliptical polarization consists of unequal amplitudes of linearly polarized components, and the electrical fields of the two polarizations oscillate at the same frequency but have some constant phase difference.

Light of arbitrary polarization can be represented by four numbers known as the Stokes parameters, I , Q, U , and V . I refers to the irradiance or intensity of the

light; the parameters Q, U , and V represent the extent of horizontal linear, 45 deg linear, and circular polarization, respectively.135,160,443–448

In polarimetry, the Stokes vector S of a light beam is constructed based on six flux measurements obtained with different polarization analyzers in front of the detector:

VIR − IL

where IH , IV , I+45◦ , I−45◦ , IR , and IL are the light intensities measured with a horizontal linear polarizer, a vertical linear polarizer, a +45◦ linear polarizer, a −45◦ linear polarizer, a right circular analyzer, and a left circular analyzer in front of the detector, respectively. Because of the relationship IH + IV = I+45◦ + I−45◦ = IR + IL = I , where I is the intensity of the light beam measured without any analyzer in front of the detector, a Stokes vector can be determined by four independent measurements, for example, IH , IV , I+45◦ , and IR :

From the Stokes vector, the degree of polarization (DOP), the degree of linear polarization (DOLP), and the degree of circular polarization (DOCP) are derived as

Q2 + U 2

DOLP = , (1.61)

I

If the DOP of a light field remains at unity after transformation by an optical system, this system is nondepolarizing; otherwise, the system is depolarizing.

The values of the normalized Stokes parameters, which correspond to a certain type of polarization, are presented in Table 1.2.

The Mueller matrix M of a sample transforms an incident Stokes vector Sin into the corresponding output Stokes vector Sout as

Obviously, the output Stokes vector varies with the state of the incident beam, but the Mueller matrix is determined only by the sample and the optical path. Conversely, the Mueller matrix can fully characterize the optical polarization properties of the sample. The Mueller matrix can be experimentally obtained from measurements with different combinations of source polarizers and detection analyzers. In the most general cases, a 4 × 4 Mueller matrix has 16 independent elements; therefore, at least 16 independent measurements must be acquired to determine a full Mueller matrix.

The normalized Stokes vectors for the four incident polarization states, H , V , +45 deg, and R, are, respectively,

where H , V , +45◦, and R, represent horizontal linear polarization, vertical linear polarization, +45 deg linear polarization, and right circular polarization, respectively. We may express the 4 × 4 Mueller matrix as M = [M1 M2 M3 M4], where M1, M2, M3, and M4 are four column vectors of four elements each. The four output Stokes vectors that correspond to the four incident polarization states, H , V , +45 deg, and R, are denoted, respectively, by SH o, SV o, S+45◦o, and SRo. These four output Stokes vectors are experimentally measured based on Eq. (1.60) and can be expressed as

S = MS = M + M

Ro Ri 1 4

The Mueller matrix can be calculated from the four output Stokes vectors as449

In other words, at least four independent Stokes vectors must be measured to determine a full Mueller matrix, and each Stokes vector requires four independent intensity measurements with different analyzers.

1.4.5 Light interaction with a random single scattering media

In terms of the electrical-field components, the Stokes parameters from Eqs. (1.59) and (1.60) are given by

V = i(E E − E E )

All Stokes parameters have the same dimension—energy per unit area per unit time per unit wavelength. For an elementary monochromatic plane or spherical electromagnetic wave,145

For an arbitrary light beam, as in the case of a partially polarized quasimonochromatic light that is due to the fundamental property of additivity, the Stokes parameters for the mixture of the elementary waves are sums of the re-

spective Stokes parameters of these waves. Equation (1.67a) is replaced by the inequality145,148

The degree of polarization (DOP), the degree of linear polarization (DOLP), and the degree of circular polarization (DOCP) for the incident and scattered light are defined by Eq. (1.61). In particular, for the DOLP (PL) and the DOCP (PC ) of the scattered light, we have

In the far field, the polarization of the scattered light is described by the Stokes vector Ss connected with the Stokes vector of the incident light Si [see Eq. (1.62)]148

The structure of the LSM further simplifies for spherically symmetric particles, which are homogeneous or radially inhomogeneous (composed of isotropic materials with a refractive index that depends only on the distance from the particle center), because in this case145

The phase function, i.e., the M11 element, for scattering symmetric relative to the direction of the incident wave depends only on the scattering angle θ and

which corresponds to assumption of random distribution of scatterers in a medium. The scattering anisotropy parameter (mean cosine of the scattering angle θ) or the asymmetry parameter of the phase function [see Eq. (1.16)] is expressed now

as

π

g ≡ cos θ = 2π M11(θ) cos θ sin θdθ. (1.77)

0

If a particle is small with respect to the wavelength of the incident light, its scattering can be described as the reemission of a single dipole. This Rayleigh theory is applicable under the condition that m(2πa/λ) 1, where m is the relative

refractive index of the scatterers, (2πa/λ) is the size parameter, a is the radius of the particle, and λ is the wavelength of the incident light in a medium.148 For the

visible and NIR light and scatterers with a typical (for biological tissue) refractive index relative to the ground matter m = 1.05–1.11, the maximum particle radius must be about 12–14 nm for Rayleigh theory to remain valid. For this theory, the scattered irradiance is inversely proportional to λ4 and increases as a6; the angular distribution of the scattered light is isotropic.

The Rayleigh-Gans or Rayleigh-Debye theory addresses the problem of calculating the scattering by a special class of arbitrary shaped particles. It requires

|m − 1| 1 and (2πa /λ)|m − 1| 1, where a is the largest dimension of the particle.129,145,146,149 These conditions mean that the electrical field inside the par-

ticle must be close to that of the incident field and that the particle can be viewed

as a collection of independent dipoles that are all exposed to the same incident field. A biological cell might be modeled as a sphere of cytoplasm with a higher refractive index (ncp = 1.37) relative to that of the surrounding water medium (nis = 1.35); then m = 1.015, and for the NIR light, this theory is valid for particle dimensions up to a = 0.8–1.0 μm. This approximation has been applied extensively to calculations of light scattering from suspensions of bacteria.149 It can be applicable for describing light scattering from cell components (mitochon-

dria, lysosomes, peroxisomes, etc.) in tissues due to their small dimensions and

refraction.58,96,150–153,163,166

For describing the forward scattering caused by large particles (on the order of 10 μm), the Fraunhofer diffraction approximation is useful.149 According to this theory, the scattered light has the same polarization as that of the incident light and the scatterer pattern is independent of the refractive index of the object. For small scattering angles, the Fraunhofer diffraction approximation can accurately represent the change in irradiance as a function of particle size. That is why this approach is applicable in laser flow cytometry. The structures in a biological cell, such as cell membrane, nuclear texture, and granules in the cytoplasm, can be detected by variations in optical density. An optical Fourier transform of the diffraction pattern can be performed by a lens. Spatial variations in optical density in the object plane are converted by a Fourier transform into spatial frequency variations in the Fourier transform plane in the rear focal plane of the lens.149 If the optical density changes slowly across the object, the Fourier transform places most of the scattered light near zero angles (low spatial frequency) in the Fourier transform plane. This is a good model of a cell with clear cytoplasm (constant optical density). If the optical density changes rapidly across the object, the Fourier transform moves more of the energy to larger scattering angles (higher spatial frequency) in the Fourier transform plane. This is a good model of a cell with highly granular cytoplasm (rapid changes in optical density across the cytoplasm). It was shown that the transforms of abnormal cells have significantly high spatial frequency compared with the transforms of normal cells, in particular single cells in cervical smears. Fourier optical microscopes were developed for such studies. The technique is applicable for a positive photographic transparency of the cell, single cells on slides, and cells in flows.149

Mie or Lorenz-Mie scattering theory is an exact solution of Maxwell’s electromagnetic field equations for a homogeneous sphere.148,160 In the general case,

light scattered at a particle becomes elliptically polarized. For spherically symmetric particles of an optically inactive material, the Mueller scattering matrix is given

by Eqs. (1.72) and (1.75). Mie theory has been extended to arbitrary coated spheres and to arbitrary cylinders.145,146,149 In the Mie theory, the electromagnetic fields of

the incident, internal, and scattered waves are each expanded in a series.160 A lin-

ear transformation can be made between the fields in each of the regions. This approach can also be used for nonspherical objects such as spheroids.145,146 The

linear transformation is called the transition matrix (T-matrix). The T-matrix for spherical particles is diagonal.

Thus far, Stokes vectors have been defined for the case of a monochromatic plane wave, and the Mueller matrix for single scattering. These concepts have been generalized to more complicated situations. The Stokes vector was defined for a quasi-monochromatic wave.181 Then in the case of partially polarized light, the inequality, described by Eq. (1.67b), is valid.148

When Mueller matrices from an ensemble of particles differing in size, orientation, morphology, or optical properties are added incoherently, six of the above-

mentioned equalities became inequalities.381 For an ensemble of interacting particles in the single scattering approximation,5,6,10,442 LSM elements have the form

where Mij0 are the LSM elements of an isolated particle, N is the number of scatterers, and Fint(θ) is the interference term, taking into account the spatial correlation of particles. Note that the normalized elements (Mij /M11) in a monodisperse system weakly depend on whether account is taken of the spatial correlation of scatterers, and this ratio is close to that for isolated particles.442

1.4.6 Vector radiative transfer equation

As it was already shown, the majority of tissues are turbid media showing a strong scattering and much less absorption (up to two orders less than scattering in the visible and NIR ranges). Moreover, in their natural state (nonsliced), tissues are

rather thick. Therefore, multiple scattering is a specific feature of a wide class of

tissues.1–3,6,24,31,129,130

Polarization effects at light propagation through various multiply scatter-

ing media, including tissues, are fully described by the vector radiative transfer equation (RTE).59,145,146,188,344–366 The radiative transfer theory (RTT) originated

as a phenomenological approach based on considering the transport of energy through a medium filled with a large number of particles and ensuring energy conservation.182–185 This medium, composed of discrete, sparsely, and randomly distributed particles, is treated as continuous and locally homogeneous. Discussed above, the concept of single scattering and absorption by an individual particle is thus replaced in this subsection by the concept of single scattering and absorption by a small homogeneous volume element. In the framework of the RTT, the scattering and absorption of the small volume element follow from the Maxwell equations and are given by the incoherent sums of the respective characteristics of the constituent particles; the result of scattering is not the transformation of a plane incident wave into a spherical scattered wave but the transformation of the specific

intensity vector (Stokes) of the incident light into the specific intensity vector of the scattered light.146,188

For macroscopically isotropic and symmetric plane-parallel scattering media, the vector radiative transfer equation (VRTE) can be substantially simplified as

follows:146,188

dS(¯r, ϑ, ϕ) = −S(¯r, ϑ, ϕ) dτ(¯r)

+(¯r) +1 d(cos ϑ ) 2π dϕ Z(¯r, ϑ, ϑ , ϕ − ϕ )S(¯r, ϑ , ϕ ),

(1.79)

where S is the Stokes vector defined by Eq. (1.59); ¯r is the position vector; ϑ, ϕ are the angles characterizing incident direction, respectively, the polar (zenith) and the azimuth angles;

is the optical-path-length element, ρ is the local particle-number density, σext is the local ensemble-averaged extinction coefficient, ds is the path-length element measured along the unit vector of the direction of light propagation; is the single scattering albedo; ϑ , ϕ are the angles characterizing scattering direction, respectively, the polar (zenith) and the azimuth angles; Z is the normalized phase matrix

where M(θ) is the single scattering Mueller matrix, defined by Eq. (1.62); θ is the scattering angle, and R(φ) is the Stokes rotation matrix for angle φ:

Every Stokes vector and Mueller matrix are associated with a specific reference plane and coordinates. In the Mie theory, the Mueller matrix of a single scattering event is defined in the scattering plane that is formed by the incident light vector and the scattered light vector (see Fig. 1.23). For a general coordinate system associated with this scattering plane, the z-axis is along the direction of photon propagation. The x-axis is within the reference plane and is perpendicular to the z-axis. The y-axis is perpendicular to both the z-axis and the reference plane.

There is a local coordinate system associated with each incident photon packet, and its Stokes vector Sin is associated with this local coordinate system. The local coordinate system of the photon before scattering is (x, y, z). After the scattering event, the photon propagates along the z -axis with θ as the polar scattering angle and ϕ as the azimuth angle. The scattering plane is formed by the z-axis and the z -axis, which is the new reference plane.

To calculate the Stokes vector of the scattered light, Eq. (1.59) is used. Because the Mueller matrix of the scattering event defined in the reference plane [see

Eq. (1.62)], we need first to transform the Stokes vector of the incident light to the coordinate system associated with the reference plane. This transformation can be done by rotating the local coordinate system (x, y, z) by φ about the z-axis, where the rotation matrix is defined by Eq. (1.82). The new Stokes vector is obtained by

The local coordinate system of the photon packet is tracked in the process. The transformation can be divided into two steps. The first step is rotating the (x, y, z) system by φ about the z-axis, and the second step is to rotate the coordinate by θ about the rotated y-axis to get (x , y , z ). After the transformation, the z -axis is aligned with the new light vector. The transformation matrix is

After a photon packet passes through the turbid medium, its Stokes vector is recorded and accumulated. The local coordinate system is tracked in the simulation. In order to record the Stokes vector, the local coordinate system of each photon packet needs to be transformed into the laboratory coordinate system. In the laboratory coordinate system (e1, e2, e3), the local photon coordinate can be written as

To transform the photon Stokes vector from the local coordinate system into the laboratory coordinate system, the local coordinate system is rotated about its z-axis so that the new x-axis lies within the (e2, e3) plane in the laboratory coordinate. The rotation angle is

The rotation matrix and the new Stokes vector can be obtained from Eqs. (1.82) and (1.83).

The phase matrix, Eq. (1.81), links the Stokes vectors of the incident and scattered beams, specified relative to their respective meridional planes. To compute the Stokes vector of the scattered beam with respect to its meridional plane, one must calculate the Stokes vector of the incident beam with respect to the scattering plane, multiply it by the scattering matrix (to obtain the Stokes vector of the scattered beam with respect to the scattering plane), and then compute the Stokes vector of the scattered beam with respect to its meridional plane. Such a procedure involves two rotations of the reference plane: = −φ; = π − φ and = π + φ;

and = φ. The scattering angle θ and the angles and are expressed via the polar and the azimuth incident and scattering angles:

cos θ = cos ϑ cos ϑ + sin ϑ sin ϑ cos(ϕ − ϕ),

The first term on the right-hand side of the VRTE [Eq. (1.79)] describes the change in the specific intensity vector over the distance ds caused by extinction and dichroism, the second term describes the contribution of light illuminating a small volume element centered at ¯r from all incident directions and scattered into the chosen direction. For real systems, the form of the VRTE tends to be rather complex and often intractable. Therefore, a wide range of analytical and numerical techniques have been developed to solve the VRTE. Because of the important property of the normalized phase matrix, Eq. (1.81), being dependent on the dif-

ference of the azimuthal angles of the scattering and incident directions rather than on their specific values,146,188 an efficient analytical treatment of the azimuthal

dependence of the multiply scattered light, using a Fourier decomposition of the VRTE, is possible. The following techniques and their combinations can be used to solve the VRTE: transfer matrix method, the singular eigenfunction method, the perturbation method, the small-angle approximation, the adding-doubling method,

the matrix operator method, the invariant embedding method, and the Monte Carlo

method.129,138,145,146,187,188,344–368

When the medium is illuminated by unpolarized light and/or only the intensity of multiply scattered light needs to be computed, the VRTE can be replaced by its approximate scalar counterpart. In that case, in Eq. (1.79), the Stokes vector is replaced by its first element (i.e., radiance) [see Eq. (1.9)] and the normalized phase matrix by its (1, 1) element (i.e., the phase function) [see Eq. (1.13)]. The scalar approximation gives poor accuracy when the size of the scattering

particles is much smaller than the wavelength, but provides acceptable results for particles comparable to and larger than the wavelength.146,184 There is ample literature1,3,15,129,130,185,188–190 on the analytical and numerical solutions of the

scalar RTE, Eq. (1.9).

1.4.7 Monte Carlo simulation

The Monte Carlo (MC) method, being widely used for the numerical solution of the RTT equation368–370 in different fields (astrophysics, atmosphere and ocean op-

tics, etc.), appears to be especially promising for the solution of direct and inverse

radiation transfer problems for media with arbitrary configurations and boundary

conditions, in particular for the purposes of the medical polarization optical tomography and spectroscopy.1,3,15,41,129,213,215,349,357,361–367 The method is based on the

numerical simulation of photon transport in scattering media. Random migrations of photons inside a sample can be traced from their input until absorption or output occur.

The straightforward simulation using the MC method has the following advantages: (1) one can employ any scattering matrix; (2) there are no obstacles for the use of strongly forward directed phase functions or experimental single scattering matrices; (3) the polarization calculation takes only a twofold increase in computation time over that needed for the evaluation of intensity; (4) any reasonable number of detectors can be accounted for without noticeable increase of the computation time; (5) there are no difficulties in determining the radiation parameters inside the medium; (6) it is possible to model media with complex geometry where radiance depends not only on the optical depth, but also on the transverse coordinates.

The liability of the obtained results to statistical variations on the order of a few percent at an acceptable computation time is the main disadvantage of the MC technique. For a twofold increase of the accuracy, one needs a fourfold increase in the computation time. The MC method is also impractical for great optical depths (τ > 100).