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Ecuación de Transferencia Radiativa

Storyboard

El transporte de fotones por materia (incluido tejido biologico) puede ser modelado mediante la ecuación de transporte radiativo (Radiative transfer equation - RTE).

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Geometrias

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Radiance function of the spectral radiance

Equation

The spectral radiance L_{\Omega,
u} is the energy per area of the photons of frequency
u emitted at a solid angle d\Omega.

If the spectral radiance is integrated in the frequency, the total radiance is obtained:

$L_i(\vec{x},\hat{n},t)=\displaystyle\int d\nu L_{i,\nu}(\vec{x},\hat{n},t)$

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Proyección de D3 a D2

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Radiant flow

Equation

The integration of the radiance L on the solid angle d\Omega gives us the radiative flux \Phi

$\Phi(\vec{x},t)=\displaystyle\int_{4\pi} L(\vec{x},\hat{n},t)d\Omega=\sum_iL_i(\vec{x},\hat{n},t)$

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Definición de Bordes en D2Q7

Image

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Radiance as function of the radiative flow

Equation

Radiance is the derivative of radiative flux at the angle and projected surface section S\cos\theta

$L_i(\vec{x},t)=\displaystyle\frac{\partial^2\Phi_i(\vec{x},t)}{\partial\Omega\partial S\cos\theta}$

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Radiant flow in function of the energy

Equation

The radiative flux is the radiative energy that by time is irradiated:

$\Phi(\vec{x},t)=\displaystyle\frac{\partial Q}{\partial t}$

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Radiative intensity

Equation

The radiative intensity is the radiative flux per element of solid angle:

$I_{\Omega}=\displaystyle\frac{\partial\Phi}{\partial\Omega}$

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Radiative transport equation (RTE)

Equation

The photon transport equation is

$\displaystyle\frac{1}{c}\displaystyle\frac{\partial}{\partial t}L(\vec{x},\hat{n},t)+\hat{n}\cdot\nabla L(\vec{x},\hat{n},t)=-\mu_tL(\vec{x},\hat{n},t)+\mu_s\int_{4\pi}L(\vec{x},\hat{n}_h,t)P(\hat{n}_h,\hat{n})d\Omega_h+S(\vec{x},\hat{n},t)$

where \mu_t is the absorption coefficient and scattering, c the velocity of light, P(\hat{n}',\hat{n}) is the phase function that gives the probability that a photon traveling in the direction \hat{n} is deflected in the direction \hat{n}'.

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