Análisis del Problema 1: Empaquetamiento a 13556’ MD

3.2.1 Scattering mechanisms

Generally biological tissue is scatter-dominated at NIR wavelengths, so that the main intensity loss of a collimated incident beam is due to scattering. With typical mean free scattering pathlengths (distance between scattering events) of the order of 10'^ mm any

transillumination measurement with a source-detector separation of 1 mm and above must take into account the effects of multiple scattering, while the unscattered component will be negligible in most cases.

The investigation of single scattering mechanisms is essential if one is to understand the processes involved in multiple scattering systems. In discrete photon models, for example, multiple scattering in optical dense media can be simulated using only the single scattering parameters.

When a plane wave interacts with a single particle, a part of its energy will be absorbed, and another part scattered. The intensity of the scattered wave generally depends on the direction ? , of the incident wave and the direction ^ of the scattered wave, and is characterised by the dijferential scattering cross section a given by [Ishimaru?8, BohrenS3]

<y(e^,e^,) = l i m _ _ (3*^) r-ioo 1.

where 7^ and 7, are the scattered and incident intensities. Frequently a dimensionless

phase function f{e ,e ,) is used instead of a:

(3-8)

where is the scattering cross section obtained by integrating a over all angles:

(3.9)

Ak

The phase function /( ? , ? ,) is normalised by J/(?^ , ? /)d ? = 1 and describes the ratio

between the intensity scattered into the direction e^ and the intensity incident in the direction?,. In the photon picture, the phase function may be interpreted as the

probability density of an incident photon being scattered from direction e^, into direction by the scattering particle.

The total cross section or attenuation cross section a, which determines the attenuation of the wave due to absorption and scattering is given by the sum

a = a + a (3.10)

In analogy to the absorption coefficient (3.3) the scattering and the total attenuation coefficients can be defined as the product of the particle density p and the corresponding cross section:

(3.11)

and in the case of a medium consisting of a mixture of scattering compounds depending of wavelength, position and time:

(3.12)

A very useful tool to describe the scattering of electromagnetic waves by small particles is the Mie theory. This theory provides an exact solution of the Maxwell equations for the scattering of a plane electromagnetic wave by

an isotropic homogeneous

spherical particle of arbitrary size. The most important result in the context of this work is the non-isotropic angular distribution of the amplitude of the scattered wave which depends on the size of the scatterer. Figure 3.1

■§ (0 0 1 •2 3 4 180 120 150 90 30 60 0

Figure 3.1. Angular intensity distribution for the unpolarised field scattered by a sphere of radius 1 fim, calculated by Mie theory. Tlie wavelength is 800 nm.

shows the angular distribution of the intensity of the field scattered by a spherical particle of radius a = \ |im and refractive index n = 1.4, for a wavelength of À = 800 nm. The scattered wave is highly peaked in the forward direction. The small lobes for high scattering angles (note the logarithmic scale) are caused by interference effects. The forward bias increases for larger particles.

When the particle size is much smaller than the wavelength of the incident wave then Mie theory turns into Rayleigh scattering theory. For an unpolarised incident wave of a fixed wavelength Rayleigh scattering gives the intensity of the scattered wave proportional to (1 + cos^ 0), as a function of the scattering angle 0, i.e. symmetric in the forward and backward direction. Another fundamental result of the Rayleigh theory is the fact that the intensity of the scattered wave is proportional to i.e. the scattering cross section of the particle decreases strongly with increasing wavelength.

A detailed discussion of the theories developed to describe the scattering of electromagnetic waves by small particles is beyond the scope of this work, and the reader is referred to one of the numerous textbooks on this topic. See for example Bohren and Huffmann [BohrenS3].

3.2.2 Forward scattering

If scattering is considered to be axially symmetric relative to the original propagation direction, then for a random scattering medium the phase function / will only depend upon the scattering angle 0 between the forward (i.e. unscattered) direction and the direction of the scattered beam: /( ? , e^,) = /(? • ? ,) = f{u) where u = cos 0. In this case the scattering characteristic is often described by the mean cosine of the scattering angle, / :

/ = Jm/(m) dw (3.13)

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equally distributed over all angles; / > 0 indicates scattering predominantly into the

forward direction 0 < 90°, while / < 0 indicates scattering into the backward direction

0 > 90°. / = 1 is the extreme case where all light is scattered into the direction 0 = 0°.

For the scattering distribution of Figure 3.1 / = 0.902 is found.

Light scattering in biological tissue is generally forward concentrated, with typical anisotropy factors of / ~ 0.9, therefore scattering particle sizes of the order of 1 qm as was assumed for Figure 3.1 is a plausible conjecture. However, it is not the intention of this work to derive the scattering and absorption parameters of tissue on a theoretical basis, and we resort to the experimentally acquired data available in the literature.