absortion curso.pdf

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1. Absorption

1.1. Some biological chromophores

1.2. Definition and units of absorption coefficient µa [cm-1]

1.3. Example: bilirubin

1.4. Absorption spectra for biological tissues

2. Scattering

2.1. Some biological scatterers

2.2. Definition and units of scattering coefficient µs [cm-1]

2.3. Definition of anisotropy g [dimensionless]

2.3.1. Scattering functions

2.3.2. Isotropic scattering function

2.3.3. Henyey-Greenstein scattering function

2.4. Reduced scattering coefficient µs ' = µs(1 - g) [cm-1]

2.5. Mie theory model for scattering

2.5.1. Brief overview of Mie theory math

2.5.2. Example calculation of angular scattering pattern

2.5.3. Example calculation of anisotropy 2.5.4. Example calculation of scattering cross section and coefficient

2.5.5. Scattering versus wavelength for 3 particle sizes

2.6. Mie scattering from cellular structure of soft tissues

2.7. Mie scattering from collagen fibers of dermis

Absorption

A simple analogy for the absorption of light by molecules is placing two identical bells side by side.

When one bell is rung, the other will sympathetically ring at the same frequency due to transfer of

energy from the struck bell. The resonance of the second bell matches the frequency of the first

ringing bell and hence the second bell accepts energy from the struck bell.

Photons are electromagnetic waves with a particular frequency. Molecules are a system with charge

separation (negative electron field and positive nucleus). The state of the molecular charge separation

can change in a quantized fashion by "absorbing" the energy of a photon. The photon frequency mustmatch the "frequency" associated with the molecule's energy transition in order for energy transfer

to occur. The relation between frequency and energy is

Energy = h(frequency) = hc/(wavelength)where Energy is in [J], photon frequency is in [cycles per s] or [s-1], photon wavelength is in [m], c

is the speed of light in vacuo (c = 3.0x108 [m/s]), and h is Planck's constant (h = 6.62618x10-34 [J s]).

Unlike the bell analogy, photon absorption occurs as a quantum event, an all or none

phenomenon. Example: absorption of 514 nm photon from an argon ion laser by a hemoglobinmolecule.

In biomedical optics, absorption of photons is a most important event:

Absorption is the primary event that allows a laser or other light source to cause a potentially therapeutic (or damaging) effect on a tissue. Without absorption, there is no

energy transfer to the tissue and the tissue is left unaffected by the light.

Absorption of light provides a diagnostic role such as the spectroscopy of a tissue. Absorptioncan provide a clue as to the chemical composition of a tissue, and serve as a mechanism of

optical contrast during imaging. Absorption is used for both spectroscopic and imaging

applications.

Example: Heme absorption of photon

For example, a 514 nm photon from an argon ion laser when absorbed by a hemoglobinmolecule will transfer energy to the hemoglobin molecule:

hc/(0.514x10-1 [m]) = 3.86x10-19 [J]

Doesn't sound like much energy, does it? But consider the energy from the perspective of theabsorbing chromophore. The photon is absorbed by the heme chromophore within

the hemoglobin protein. The heme choromophore is roughly 1 nm is size.

After the heme chromophore absorbs one photon, the jump in energy density is roughly thephoton energy divided by a nm3 volume:

(3.86x10-19

[J])/(10-7

[cm])3

= 387 [J/cm3

]

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For comparison, the energy density of boiling water is 418 J/cm3. So one photon carries quite

a lot of energy from the perspective of a small molecule!

1.1 Some biological chromophores Molecules that absorb light are called chromophores. There are two major types of choromphores:

electronic transitions

vibrational transitions

Electronic transitions There are many biological molecules which can absorb light via electronic transitions. Such

transitions are relatively energetic and hence are associated with absorption of ultraviolet, visible and

near-infrared wavelengths. The molecules generally have a string of double bonds whose pi-orbitalelectrons act similar to the electrons in a metal in that they collectively behave as a small antenna

which can "receive" the electromagnetic wave of a passing photon. If the resonance of the pi-orbital

structure matches the photon's wavelength then photon absorption is possible.In early biological evolution, the pyrrole molecule was a chromphore which could absorb sunlight

which enabled subsequent synthetic reactions that produced biological polymers and other proto-

metabolic products. Combining four pyrroles into a tetrapyrrole ring (porphyrin) yielded an efficient

chromophore for collecting solar photons. Chlorophyll is such a porphyrin. Hemoglobin, vitaminB12, cytochrome C, and P450 are also examples of porphyrins in biology. The figure lists some

common biological chromophores and shows some of their structures. Also see the

website spectra which is a compilation of chromophores, most absorbing in the ultraviolet andvisible.

Vibrational transitions The field of infrared spectroscopy studies the variety of bonds which can resonantly vibrate or twist

in response to infrared wavelengths and thereby absorb such photons. Perhaps the most dominantchromophore in biology which absorbs via vibrational transitions is water. In the infrared, the

absorption of water is the strongest contributor to tissue absorption.

Some vibrational frequencies

bond cycles/cm, wavelength, = 1/

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C-H stretch 2850-2960 [cm-1] 3.378-3.509 [µm]

C-H bend 1340-1465 6.826-7.462

C-C stretch,bend 700-1250 8.000-14.29

C=C stretch 1620-1680 5.952-6.173

C=C stretch 2100-2260 4.425-4.762

CO32- 1410-1450 6.897-7.092

NO3- 1350-1420 7.042-7.407

NO2- 1230-1250 8.000-8.130

SO42- 1080-1130 8.850-9.259

O-H stretch 3590-3650 2.740-2.786

C=O stretch 1640-1780 5.618-6.098

N-H 3200-3500 2.857-3.125

ref: PW Atkins, "Physical Chemistry," p. 576, W.H. Freeman and Co., 1978.1.2 Definition and units of absorption coefficient µa [cm-1] Consider a chromophore idealized as a sphere with a particular geometrical size. Consider that thissphere blocks incident light and casts a shadow, which constitutes absorption. This description is of

course an incorrect and schematicized version of the real situation. However, it does provide a simple

concept which captures the essence of the absorption coefficient, the parameter we use to describe

the effectiveness of absorption.The size of the absorption shadow is called the effective cross-section ( a [cm2]) and can be smaller

or larger than the geometrical size of the chromophore (A [cm2]), related by the proportionality

constant called the absorption efficiency Qa [dimensionless]:

The absorption coefficient µa [cm-1] describes a medium containing many chromophores at a

concentration described as a volume density a [cm3]. The absorption coefficient is essentially the

cross-sectional area per unit volume of medium.

Experimentally, the units [cm-1] for µa are inverse length, such that the product µaL is dimensionless,

where L [cm] is a photon's pathlength of travel through the medium. The probability of survival (or

transmission T) of the photon after a pathlength L is:

This expression for survival holds true regardless of whether the photon path is a straight line or ahighly tortuous path due to multiple scattering in an optically turbid medium.

1.3 Example: Bilirubin

The bilirubin molecule is a chromophore encountered when a newborn infant suffers from "jaundice",

a syndrome in which the skin presents a yellow color. Bilirubin is a breakdown product ofhemoglobin. Often there is significant hemolysis of red blood cells during child birth contributing to

a transient bilirubin load. Normally, bilirubin binds to the serum protein albumin and is carried to the

liver where enzymes convert it into a water-soluble form which is removed from the blood into the


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bile. But in newborns who have not yet developed sufficient enzymes to accomplish this task, the

bilirubin accumulates in the blood, exceeds the holding capacity of the albumin, and spills into theskin to give the yellow skin color, and into the brain to cause irreversible brain damage (kernicterus).

See article on optically monitoring bilirubinemia.

Let's approximate the values of A, a, µa, Qa, and a for bilirubin.

The structure of bilirubin is shown. The diameter isapproximately 1 nm. So the geometrical area is A =

4.5x10-15 cm2.

At 460 nm, the extinction coefficient of bilirubin is =

53,846 [cm-1M-1] (see bilirubin spectrum).

The extinction coefficient [cm-1M-1] quoted in theliterature is based on spectrometer measurementsreported as T = 10- CL where C is concentration [M] and

L is pathlength [cm]. Therefore,µa = C ln(10)

A typical jaundiced neonate might have a bilirub in

concentration of 10 mg/dL, or (0.100 g/liter)/(574.65g/mole) = 0.17x10-3 M. In such a case, the bilirub in

absorption coefficient at 460 nm is roughly

µa = C ln(10) = (53846 [cm-1M-1])(0.17x10-3 M)(2.3)

= 21 cm-1.

The concentration C is equivalent to

a =(0.17x10-3 [moles/liter])(6x1023 [mole-1])/(1000

cm3/liter] = 1.02x1017 [cm-3]

The efficiency of absorption is estimated:Qa = µa/( aA) = (21 [cm-1])/((1.02x1017 [cm-3])(A =

4.5x10-15 [cm2])) = 0.046

The effective cross-section is a = QaA = (0.046)(4.5x10-15 [cm2]) = 2.1x10-16 [cm2]. Bilirubin's effective cross-

sectional diameter is sqrt(.046) or 21% the size of itsgeometrical diameter.

Keep in mind that the wavelength of blue light (460 nm) is about 460-fold greater than the diameter

of the bilirubin chromophore. So collection of the electromagnetic wave of a photon by the "antenna"of bilirubin is analogous to a how small radio antenna collects the electromagnetic wave from a radio

station (a 1000 MHz radio frequency -> 300 m wavelength). The concept of a "shadow" cast by anopaque chromophore is merely a memory device to remember the definitions.

1.4 Absorption spectra for biological tissues

The figure below shows the primary absorption spectra of biological tissues. Also shown are the

absorption coefficients at some typical laser wavelengths.

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There are several major contributors to the absorption spectrum:

In the ultraviolet, the absorption increases with shorter wavelength due to protein, DNA andother molecules.

In the infrared, the absorption increases with longer wavelengths due to tissue water content.Scaling the pure water absorption by 75% mimics a typical tissue with 75% water content. In the red to near-infrared (NIR), absorption is miminal. This region is called the diagnostic

and therapeutic window (originally by John Parrish and Rox Anderson).

Whole blood is a strong absorber in the red-NIR regime, but because the volume fraction of

blood is a few percent in tissues, the average absorption coefficient that affects light transportis moderate. However, when photons strike a blood vessel they encounter the full strong

absorption of whole blood. Hence, local absorption properties govern light-tissue interactions,

and average absorption properties govern light transport. Melanosomes are also strong absorbers. However, their volume fraction in the epidermis may

be quite low, perhaps several percent. So again, the local interaction of light with themelanosomes is strong, but the melanosome contribution to the average absorption coefficientmay modestly affect light transport.

2 Scattering

Scattering of light occurs in media which contains fluctuations in the refractive index n, whether such

fluctuations are discrete particles or more continuous variations in n.

In biomedical optics, scattering of photons is an important event: Scattering provides feedback during therapy. For example, during laser coagulation of tissues,

the onset of scattering is an observable endpoint that correlates with a desired therapeutic goal.

Scattering also strongly affects the dosimetry of light during therapeutic procedures thatdepend on absorption. The scattering affects "where" the absorption will occur.

Scattering has diagnostic value. Scattering depends on the ultrastructure of a tissue, eg., the

density of lipid membranes in the cells, the size of nuclei, the presence of collagen fibers, thestatus of hydration in the tissue, etc. Whether one measures the wavelength dependence of

scattering, the polarization dependence of scattering, the angular dependence of scattering, the

scattering of coherent light, scattering measurements are an important diagnostic tool.Scattering is used for both spectroscopic and imaging applications.

2.1 Some biological scatterers

The light scattered by a tissue has interacted with the ultrastructure of the tissue. Tissue ultrastructure

extends from membranes to membrane aggregates to collagen fibers to nuclei to cells. Photons aremost strongly scattered by those structures whose size matches the photon wavelength.

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Scattering of light by structures on the same size scale as the photon wavelength is described by Mie

theory. Scattering of light by structures much smaller than the photon wavelength is called theRayleigh limit of Mie scattering, or simply Rayleigh scattering. The figure designates the size range

of tissue ultrastructure which affects visible and infrared light by Mie and Rayleigh scattering.

Here are some examples of structures which scatter light (click on figure to expand):

Mitochondria

Mitochondria are intracellular organelles about 1 µm in length

(variable) which are composed of many folded internal lipid

membranes called cristae, as shown in the electron micrograph at

left. The basic lipid bilayer membrane is about 9 nm in width. The

refractive index mismatch between lipid and the surrounding

aqueous medium causes strong scattering of light. Folding of lipid

membranes presents larger size l ipid structures which affect longer

wavelengths of light. The density of lipid/water interfaces within the

mitochondria make them especially strong scatterers of light.

(Drawing and micrographs from A. L. Lehninger, "Biochemistry", Worth Publishers, 1970. )

Collagen fibers, fibrils, and fibril periodicity

Collagen fibers (about 2-3 µm in diameter) are composed of

bundles of smaller collagen fibrils about 0.3 µm in diameter

(variable), as shown in the electron micrograph at left. Mie

scattering from collagen fibers dominates scattering in the infrared

wavelength range.

On the ultrastructural level, fibrils are composed of

entwined tropocollagen molecules. The fibrils present a banded

pattern of striations with 70-nm periodicity due to the staggered

alignment of the tropocollagen molecules which each have anelectron-dense head group that appears dark in the electron

micrograph. The periodic fluctuations in refractive index on this

ultrastructureal level appear to contribute a Rayleigh scattering

component that dominates the visible and ultraviolet wavelength

ranges.

2.2 Definition and units of scattering coefficient

µs [cm-1]

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Consider a scattering particle idealized as a sphere with a particular geometrical size. Consider that

this sphere redirects incident photons into new directions and so prevents the forward on-axistransmission of photons, thereby casting a shadow. This process constitutes scattering. This

description is of course an oversimplified and schematicized version of the real situation. However,

it does provide a simple concept which captures the essence of the scattering coefficient, a parameteranalogous to the absorption coefficient discussed previously.

The size of the scattering shadow is called the effective cross-section ( s [cm2]) and can be smaller

or larger than the geometrical size of the scattering particle (A [cm2]), related by the proportionalityconstant called the scattering efficiency Qs [dimensionless]:

The scattering coefficient µs [cm-1] describes a medium containing many scattering particles at a

concentration described as a volume density s [cm3]. The scattering coefficient is essentially the

cross-sectional area per unit volume of medium.

Experimentally, the units [cm-1] for µs are inverse length, such that the product µsL is dimensionless,

where L [cm] is a photon's pathlength of travel through the medium. The probability of transmission

T of the photon without redirection by scattering after a pathlength L is:

2.3 Definition of anisotropy g [dimensionless]

The anistoropy, g [dimensionless], is a measure of the amount of forward direction retained after a

single scattering event. Imagine that a photon is scattered by a particle so that its trajectory is

deflected by a deflection angle , as shown in the figure below. Then the component of the newtrajectory which is aligned in the forward direction is shown in red as cos( ). On average, there is an

average deflection angle and the mean value of cos( ) is defined as the anisotropy. 13

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A scattering event causes a deflection at angle from the original forward trajectory. There is alsoan azimuthal angle of scattering, .

But it is the deflection angle which affects the amount of forward direction, cos( ), retained by the

photon.Consider an experiment in which a laser beam strikes a target such as a cylindrical cuvette containing

a dilute solution of scattering particles. The scattering pattern p( ) is measured by a detector that is

moved in a circle around the target while always facing the target. Hence the detector collects lightscattered at various deflection angles in a horizontal plane parallel to the table top on which theapparatus sits. The proper definition of anisotropy is the expectation value for cos( ):

It is common to express the definition of anisotropy in an equivalent way:

2.3.1 Scattering functions

The angular dependence of scattering is called the scattering function, p( ) which has units of [sr -1]and describes the probability of a photon scattering into a unit solid angle oriented at an

angle relative to the photons original trajectory. Note that the function depends on only on the

deflection angle and not on the azimuthal angle . Such azimuthally symmetric scattering is aspecial case, but is usually adopted when discussing scattering. However, it is possible to consider

scattering which does not exhibit azimuthal symmetry. The p( ) has historically been also called the

scattering phase function.The scattering can be described in two ways:

Plotting p( ) indicates how photons will scatter as a function of in a single plane ofobservation (source-scatterer-observer plane). This pattern is similar to the type of goniometric

scattering experiments commonly conducted. Plotting p( )2 sin indicates how photons will scatter as a function of the deflection

angle regardless of the azimuthal angle , in other words integrating over all possible in an

azimuthal ring of width d and perimeter 2 sin at some given . The p( )2 sin goes to zeroat 0° because the azimuthal ring becomes vanishingly small at 0°. This plot is related to the

total energy scattered at a given deflection angle and hence is more pertinent to the value of

anisotropy.

Figure depicts a typical forward-directed scattering pattern p( )

corresponding to an experimental goniometric measurement in a single source-scatterer-observer

plane, and p( )2 sin which integrates over all possible azimuthal angles .

2.3.2 Isotropic scattering function


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An isotropic scattering function would scatter light with equal efficiency into all possible

directions. Such a scattering function would have the form:

2.3.3 Henyey-Greenstein scattering function

enyey and Greenstein (1941) devised an expression which mimics the angular dependence of lightscattering by small particles, which they used to describe scattering of light by interstellar dust clouds.

The Henyey-Greenstein scattering function has proven to be useful in approximating the angularscattering dependence of single scattering events in biological tissues.

The Henyey-Greenstein function allows the anisotropy factor g to specify p( ) such that calculation

of the expectation value for cos( ) returns exactly the same value g. In other words, Henyey andGreenstein devised a useful identity function. The Henyey-Greenstein function is:

It is common practice to express the Henyey-Greenstein function as the function p(cos );

A series of Henyey-Greenstein functions are shown in the following figure. The forward direction

along the original photon trajectory is 0°. Scattering in the backward direction is 180°. The curve for

g = 0 has a constant value of 1/4 .

2.4 Reduced scattering coefficient

µs' = µs(1 - g) [cm-1]The reduced scattering coefficient is a lumped property incorporating the scattering coefficient

µs and the anisotropy g:

µs' = µs(1 - g) [cm-1] The purpose of µs' is to describe the diffusion of photons in a random walk of step size of 1/µs' [cm]

where each step involves isotropic scattering. Such a description is equivalent to description of

photon movement using many small steps 1/µs that each involve only a partial deflection angle if

there are many scattering events before an absorption event, i.e. , µa


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The following figure shows how many such big steps involving isotropic scattering are equivalent to

many small anisotropic steps:

2.5 Mie theory model for tissue optical properties Mie theory describes the scattering of light by particles. "Particles" here means an aggregation of

material that constitutes a region with refractive index (n p) that differs from the refractive index of

its surroundings (nmed). The dipole reradiation pattern from oscillating electrons in the molecules of

such particles superimpose to yield a strong net source of scattered radiation. Also, the reradiation patterns from all the dipoles do not cancel in all but the forward direction of the incident light as is

true for homogneous medium, but rather interfere both constructively and destructively in a radiation

pattern. Hence, particles "scatter" light in various directions with varying efficiency.Gustav Mie in 1908 published a solution to the problem of light scattering by homogeneous spherical

particles of any size. Mie's classical solution is described in terms of two parameters, nr and x:

the magnitude of refractive index mismatch between particle and medium expressed as theratio of the n for particle and medium,

nr = n p/nmed

the size of the surface of refractive index mismatch which is the "antenna" for reradiation ofelectromagnetic energy, expressed as a size parameter (x) which is the ratio of the meridional

circumference of the sphere (2 a, where radius = a) to the wavelength ( /nmed) of light in the

medium,x = 2 a/( /nmed)

A Mie theory calculation will yield the efficiency of scattering which relates the cross-sectional area

of scattering, s [cm2], to the true geometrical cross-sectional area of the particle, A = a2 [cm2]:

s = QsA

Finally, the scattering coefficient is related to the product of scatterer number density, s [cm-3], and

the cross-sectional area of scattering, s [cm2], (see definition of scattering coefficient):µs = s s

Before using Mie theory to approximate the scattering behavior of biological tissues, let's

briefly examine the Mie calculation

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illustrate the behavior of Mie scattering with some example calculations and figures

2.5.1 The math of Mie scatteringConsider a source, a spherical scattering particle, and an observer whose three positions define a

plane called the scattering plane. Incident light and scattered light can be reduced to their

components which are parallel or perpendicular to the scattering plane. As shown in the following

figure, the parallel and perpendicular components can be experimentally selected by a linear

polarization filter oriented parallel or perpendicular to the scattering plane.

The Scattering matrix describes the relationship between incident and scattered electric field

components perpendicular and parallel to the scattering plane as observed in the "far-field"

(ref: Bohren and Huffman):

The above expression simplifies in practical experiments:

The exponential term, -exp(-ik(r-z))/ikr, is a transport factor that depends on the distance

between scatterer and observer. If one measures scattered light at a contant distance r from thescatterer, eg., as a function of angle or orientation of polarization, then the transport factor

becomes a constant.

The total field (Etot) depends on the incident field (Ei), the scattered field (Es) , and the

interaction of these fields (Eint). If one observes the scattering from a position which avoidsEi, then both Ei and Eint are zero and only Es is observed.

For "far-field" observation of Es at a distance r from a particle of diameter d such that kr >>

nc2, k = 2 / , nc = d/ , the scattering elements S3 and S4 equal zero (see Eq. 4.75, Bohren andHuffman).

Practical experiments measure intensity, I = = (1/2)a2, where E = a exp(-i ), and a is

amplitude and is phase of the electric field.Hence for practical scattering measurements, the above equation simplifies to the following:

Mie theory yields two sets of descriptors of scattering:

ANGULAR SCATTERING PATTERN OF POLARIZED LIGHT Mie theory calculates the

angular dependence of the two elements, S1( ) and S2( ), of the Scattering matrix, from which the

scattered intensities of polarized light are computed (see example). The scattering pattern is also usedto calculate the anisotropy, g, of scattering by the particle.

Example of angular scattering calculation

Example of anistoropy calculation

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EFFICIENCIES OF SCATTERING AND ABSORPTION Mie theory calculates the efficiencies

Qs and Qa of scattering and absorption, respectively, such that µa = QaA and µs = QsA, where A isthe geometrical cross-sectional area a2 for a sphere of radius a.

Example of Qs and µs calculation

2.5.2 Angular patterns of Mie scattering

Consider the scattering pattern from a 0.304-µm-dia. nonabsorbing polystyrene sphere in waterirradiated by HeNe laser beam:

np = 1.5721 particle refractive index

nmed = 1.3316 medium refractive index

a = 0.152 µm, particle radius

= 0.6328 µm wavelength in vacuo

As calculated before:

nr = np/nmed = 1.5721/1.3316 = 1.1806 relative refractive index

x = 2 a/( /nmed) = (2)(3.1415)(0.152)/(0.6328/1.3316) = 2.0097 size parameter

Run the Mie theory algorithm: Mie(nr, x) ---> S1( ), S1( ) as complex numbers

Mie(1.1806, 2.0097) ---> S1.re + jS1.im, S2.re + jS2.im as functions of

To view the results, calculate the intensities of scattering for parallel and perpendicular

orientations of polarized source/detector pairs:

Ipa r = S2S2* = Re{(S2.re + jS2.im)(S2.re - jS2.im)}

Ipe r = S1S1* = Re{(S1.re + jS1.im)(S1.re - jS1.im)}which are shown in the following figures, and can be experimentally measured as described

below.

The following figures describe the experimental measurements that illustrate the angular dependence

of the scattered intensities I par and I per :

Iper Irradiate dilute solution of spheres in water with laser beam

polarization oriented perpendicular to the table. Collectscattered light as a function of angle in plane parallel to

table. Place linear polarization filter in front of

detector perpendicular to the table.

Ipar Irradiate dilute solution of spheres in water with laser beam

polarization oriented parallel to the table. Collect scattered

light as a function of angle in plane parallel to table. Place

linear polarization filter in front of detector parallel to the

table.

Example results: Polar and xy plots of scattering pattern for Ipar and Iper

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Polar plot

Click figure to enlarge

I( ) plot

Click figure to enlarge

For a randomly polarized light source, the total scattered light intensity is given by the term S11:

S11 is the first element of the so-called Mueller Matrix, a 4x4 matrix which relates an input vector of

Stokes parameters (Ii, Qi, Ui, Vi) describing a complex light source and the output vector (Is, Qs, Us,

Vs) describing the nature of the transmitted light. For randomly polarized light, S11 describes thetransport of total intensity:

Is = S11Ii

2.5.3 Mie theory calculation of anisotropy (g)

Let's use the scattering pattern calculated previously for a 0.304-µm-dia. nonabsorbing polystyrenesphere in water irradiated by HeNe laser beam to calculate the anisotropy (g) of scattering.

The definition of anisotropy guides the calculation of g using the function S11( ) which describes the

scattered intensity for randomly polarized light:

The denominator in the above equation assures proper normalization of p( ):

When numerically evaluating the above expression for g, a large number of angles need to be

calculated, typically about 200 angles, in order to achieve a value of g with precision to at least 4significant digits. For our example 0.304-µm sphere, the calculation of g based on S11( ) yields g =0.6608.

Note that the definition of g as cited in these notes assumes azimuthal symmetry, hence we have

calculated the g for randomly polarized light. Due to the symmetry presented by a spherical particle,the g refers to scattering into all azimuthal angles regardless of the linear polarization of the incident

light.

2.5.4 Example: Mie calculation of Q s and µs Consider the scattering of a HeNe laser beam by 0.304-µm-dia. nonabsorbing polystyrene spheres in

water at a concentration of 0.1% volume fraction:n p = 1.5721 particle refractive index

nmed = 1.3316 medium refractive index

a = 0.152 µm, particle radius

= 0.6328 µm wavelength in vacuo

f v = 0.001 volume fraction of particles in medium

Calculate the following parameters:

refractive index mismatch ratio, size parameter, geometrical cross-sectional area, and number density nr = n p/nmed = 1.5721/1.3316 = 1.1806

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x = 2 a/( /nmed) = (2)(3.1415)(0.152)/(0.6328/1.3316) = 2.0097

A = a2 = (3.1415)(0.152)2 = 0.0726 µm2 s = f v/((4/3) a3) = (0.001)/((4/3)(3.1415)(0.152)3) = 0.0608 µm-3

Run the Mie theory algorithm:

Mie(nr , x) ---> Qs

Mie(1.1806, 2.0097) ---> 0.1971

Calculate the scattering coefficient: s = QsA = (0.1971)(0.0726) = 0.01431 µm2 µs = s s = (0.0608)(0.01431) = 0.0009723 µm-1

µs [cm-1] = (µs [µm-1])(104 [µm/cm]) = 9.723 cm-1

2.5.5 Scattering versus wavelength for 3 particle sizes

Consider the three optical scattering properties, µs, g, and µs(1 - g), as functions of wavelength for aspherical particle with n p = 1.5721 in a medium with nmed = 1.3316 at a concentration of 0.1% volume

fraction (f v = 0.001). Assume that n p and nmed are constant versus wavelength for this example

calculation.Let the sphere be one of three sizes: 0.100 µm, 0.300 µm, and 1.00 µm.

The Mie theory calculation yields:

µs

g µs(1 - g)

Note: The true n p and nmed for polysytrene spheres and water are slightly wavelength dependent and

would deviate slightly from the above example.

2.6 Mie scattering from cellular structures

Soft tissue optics are dominated by the lipid content of the tissues. Such lipid constitutes the cellularmembranes, membrane folds, and membranous structures such as the mitochondria (about 0.5

µm). While other objects such as protein aggregates and the nucleus are also sources of scattering,

the lipid/water interface of membranes presents a strong refractive index mismatch and so plays a

major role is scattering. The following graph summarizes the lipid contents of various tissues:

Lipid contents of tissues

[ref. 1] Click figure to enlarge

Mie theory provides a simple first approximation to the scattering of soft tissues [ref. 2]. Theapproximation involves a few assumptions:

Assume the refractive index of the lipid membranes of cells is 1.46, based on the reported

1.43-1.49 range for hydrogenated fats [ref. 3].

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Assume the refractive index of the cytosol of cells is 1.35, based on the reported value for

cellular cytoplasms [ref. 1].

Assume the lipid content of soft tissue is about 1-10% (f v = 0.02-0.10). Let's choose f v = 0.02

for this example to match the value for several typical soft tissues such as lungs, spleen,

prostate, ovary, intestine, liver, arteries, to name a few.

Assume all the lipid is packaged as small spheres of various sizes whose number density

maintains a constant volume fraction f v.

Ignore the interference of scattered fields from particles which can alter the apparent

scattering properties based on isolated particles.

In the following graphs, a 2% volume fraction of lipid is packaged as spheres of one size, for a seriesof choices of sphere diameter (radius a), adjusting the number density to maintain a constant volume

fraction:

s = f v/((4/3) a3)

µs g

µs(1 - g)

This picture includes the experimental values of µ s(1 - g) for dog prostate,

il lustrating that the prostate is mimiced by a 2% volume fraction of lipid spheresin the 0.200-0.600 µm diameter range. Other soft tissues vary only a little from

this general pattern for prostate tissue.

In summary

The wavelength dependence of light scattering in soft tissues such as the prostate suggests

that cellular structures equivalent to spheres with diameters in the range of 0.200-0.600 µm contribute

most of the scattering. The magnitude of the scattering suggests that a volume fraction of 0.02 or 2%,which is roughly the reported lipid content of prostate and many other soft tissues, yields a magnitude

for µs(1 - g) which matches the magnitude of µs(1 - g) for prostate. Deviations from our assumed

values for n p based on lipid and nmed based on cytosol and from our assumed value for f v will affect

the magnitude of µs(1 - g). But in general,soft tissues with higher (or lower) lipid content will showincreased (or decreased) scattering, while the wavelength dependence of µs(1 - g) should not change

greatly.

2.7 Mie scattering from collagen fibers

Collagen fibers are strongly scattering. For example, the dermis of skin or the sclera of the eye are

tissues with high collagen fiber content. Mie theory can be used to approximate the scattering

properties of collagen on two levels: on the macroscopic level of collagen fiber bundles.

on the ultrastructural level of periodic striations in collagen fibrils.

macroscopic level of collagen fiber bundles

Collagen fibers vary from 0.1 µm-dia. fibrils to 8 µm-dia. fiber bundles. A study reported the

distribution and concentration of collagen fiber bundle diameters in 9 post-mortem neonatal skin

specimens [ref.4, Saidi et al. 1995].

The fiber diameter (d) was 2.80 ± 0.08 µm (n = 9), i.e., (mean ± SD for n specimens), with an

intraspecimen variation SD/mean = 0.43 ± 0.05.

The fiber concentration was s = 3 x106 ± 0.5 x106 cm-3. The volume fraction was f v = d2(1

cm) s/4 = 0.21 ± 0.10 (n = 9).

Mie theory can approximate collagen fiber scattering in dermis using the following assumptions:

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Assume nme d of the dermal ground substance is 1.350, based on a reported value for the n of

corneal ground substance [ref.5].

Assume the np of the collagen fiber bundles is 1.389, based on a mean dermal water content

of 65.3%, W = 0.653, (see table 9.1 of ref.1): approximately, n = 1.50 - (1.50 - 1.33)W = 1.389.

Assume one can use the cylindrical Mie theory calculation of Bohren and Huffman [appendix

C in ref.6], which assumes that the incident light is oriented perpendicular to the long axis of

the fiber cylinders.

Assume that each fiber cylinder scatters light independently, ignoring any interference

effects from closely spaced fibers.

The above assumptions allow calculation of µs(1 - g) versus wavelength for dermal collagen fiber bundles. (see "Mie" in figure below).

ultrastructural level of periodic striations in collagen fibrils

The ultrastructure of collagen fibrils presents periodic striations as was shown in the figures in

our introduction to scattering. Fibrils are composed of entwined tropocollagen molecules, presenting

a banded pattern of periodic striations (70-nm spacing) due to the staggered alignment of the

tropocollagen molecules which each have an electron-dense head group that appears dark in theelectron micrograph. The periodic fluctuations in refractive index on this ultrastructureal level appear

to contribute a Mie scattering component that dominates the visible and ultraviolet wavelengthranges. Such scattering from very small structures is called the Rayleigh limit of Mie scattering, or

simply "Rayleigh" scattering.

The following figure compares Mie theory for various size spheres withthe "Rayleigh" componentof skin scattering seen experimentally. The "Rayleigh" behavior of skin scattering is mimiced by 50-

nm-dia. spheres, n p = 1.5, nmed = 1.35, at the volume fraction of collagen in dermis, f v = 0.21. This

assignment of the "Rayleigh" scattering to the collagen ultrastructure should be regarded as only a

working hypothesis, but there is no other material in large quantity in the dermis to offer a strong

source of scattering.

Click on figure to enlarge

Combining the Mie and Rayleigh contributions for skin

The combination of the "Mie" and "Rayleigh" contributions to scattering are shown in the following

graph, along with measured skin data:

Click on figure to enlarge.

"Mie" contribution is Mie scattering from

2.8-µm-dia. cylindrical collagen fiber bundles,

np = 1.46, nmed = 1.35, f v = 0.21.

"Rayleigh" contribution is Mie scattering from

50-nm-dia. spheres mimicing the ultrastructure

associated with the periodic striations of collagen fibrils,

np = 1.50, nmed = 1.35, f v = 0.21.
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