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Analytical and numerical approximations to
the radiative transport equation for light
propagation in tissues
Arnold D. Kim
School of Natural Sciences
University of California, Merced
We seek to give an overview of the
analytical and numerical approximations
used in studying light propagation in tissues.
• The theory of radiative transport governs light
propagation in tissues.
• Typically, one make several simplifying assumptions to
the general radiative transport equation.
• We wish to become familiar with some of the more
common analytical and computational methods used in
light propagation in tissues.
Light propagation in tissues is governed by
the radiative transport equation.
This theory takes into account absorption and scattering
due to inhomogeneities.
We have made several assumptions in
applying radiative transport theory to light
propagation in tissues.
 There are no correlations between fields, so the addition
of power (not the addition of fields) holds.
• The background is homogeneous.
• There is no time dependence.
• There is no polarization.
In what follows, we discuss each of these assumptions.
There are no correlations between fields, so
the addition of power (not the addition of
fields) holds.
This is the key assumption in radiative transport theory.
For a discrete random medium, this assumption
corresponds to a medium being sufficiently dilute that
scattered fields are not correlated, but dense enough that
one can take a continuum limit.
For a continuous random medium, the random
fluctuations must be relatively small and vary on a scale
on the order of the wavelength.
The background is homogeneous.
The theory of radiative transport can be extended readily
to take into account a varying background refractive
index.
There have been several theories proposed for this
problem. G. Bal [J. Opt. Soc. Am. A 23, 1639-1644
(2006)] has reviewed the existing theories and offered a
mathematical explanation for the correct theory.
There is no time dependence.
The theory of radiative transport can be extended readily
to take into account time dependence (for a sufficiently
narrow bandwidth).
For ultra-short pulses (broad bandwidth), one may need
to consider a two-frequency radiative transport equation.
See, for example, A. Ishimaru, J. Opt. Soc. Am. 68,
1045–1050 (1978) and A. C. Fannjiang, C. R. Physique
8, 267-271 (2007).
There is no polarization.
The theory of radiative transport can be extended readily
to take into account polarization through the so-called
Stokes matrix.
Here I is a 4-vector containing the Stokes parameters (I,
Q, U and V) and S is a 4x4 scattering matrix. Polarization
is neglected very often for mathematical simplicity.
See M. Moscoso et al, J. Opt. Soc. Am. A 18, 948-960
(2001) and references therein to learn about polarization
phenomena used for light propagation in tissues.
We focus our attention on the steady-state
(continuous wave), scalar problem.
• According to radiative transport theory, there is a
diversity of data available from scattered light
measurements.
• To reach the full potential of diffuse optical tomography
and optical molecular imaging, one needs to understand
how information is contained in scattered light
measurements.
• Rather than focus on what technology can offer, we
focus on exploiting all we can from simple data sets.
Boundary conditions prescribe the light
going into the domain at the boundary
surface.
When there is an refractive index mismatch
at the boundary, we must include
transmission and reflection operators in the
boundary condition.
The scattering phase function is assumed
often to be rotationally invariant.
Henyey and Greenstein (1941) proposed a
simple, one parameter scattering phase
function.
This scattering phase function model is the one used most
commonly for light propagation in tissues.
Experimental studies have shown it to work reasonably
well at modeling scattered light measurements.
In the near-infrared portion of the spectrum,
tissues absorb light weakly and scattering
light strongly with a sharp forward peak.
W.-F. Cheong et al [IEEE J. Quant. Electron. 26, 2166-2185 (1990)]
give several results for values of the optical parameters ma, ms and g
(assuming the Henyey-Greenstein model) in tabulated form as well
as an explanation for how these estimated values were obtained.
Light is often delivered to and collected from
tissues using optical fibers.
There are a variety of optical fiber probe designs that
deliver light into tissues in different ways.
We model different modes for delivering light into tissues
through the spatial and directional dependence of the
boundary condition.
We characterize optical fiber detectors
through the numerical aperture (NA).
z
y
We model a fiber detector
measurement as
x
Optical fiber detectors measure angular
integrals of the specific intensity.
• Measurements from other detectors such as chargedcoupler device (CCD) cameras are similar even though
they measure the spatial variation over a twodimensional array.
• We can obtain directional diversity in measurements by
using different optical fibers with different numerical
apertures, but that method is not practical.
• Some experiments have measured carefully the
directional dependence of scattered light using
goniometers, but those experiments are not too
common.
We consider now several analytical
approximations that are used or are useful (we
believe) for light propagation in tissues.
•
•
•
•
PN
Diffusion
Delta-Eddington
Fokker-Planck and extensions
The PN approximation involves expanding
the specific intensity in terms of spherical
harmonics.
See S. Arridge [Inverse Problems 15, R41-R93 (1999)] for the details
of this calculation.
The PN approximation has some obvious
challenges.
The PN approximation reduces the dimensionality of the
problem substantially by projecting out the directional
dependence.
The solution of the radiative transport equation can be
discontinuous with respect to W, especially near
boundaries (recall how boundary conditions are
prescribed).
Hence, the spherical harmonics expansion would converge
slowly thereby requiring several terms for an accurate
approximation.
Otherwise, one must prescribe approximate boundary
We consider specifically the P1
approximation which yields
The diffusion approximation is substantially simpler than
the radiative transport equation and has been used
extensively to model light propagation in tissues.
We can understand better the diffusion
approximation by a different derivation.
The diffusion approximation applies to light
that has propagated deeply in an optically
thick medium.
• The specific intensity becomes nearly isotropic due to
strong multiple scattering. It is not valid near sources
and boundaries (must add boundary layer solutions).
• Many have sought extensions of the diffusion
approximation to increase its region of validity so that it
may apply to a broader range of problems.
• Regardless, most state-of-the-art optical tomography
and optical molecular imaging reconstructions are
based on the diffusion approximation.
The diffusion approximation takes into
account that tissues scattering light strongly
and absorb light weakly, but it does not take
into account explicitly that scattering is
sharply peaked in the forward direction.
As g ! 1, the diffusion length scale l* = [ms(1-g)]-1
becomes much larger than the scattering mean free
path ls = ms-1.
When these two scales separate light may scatter
multiply over a non-trivial length scale before becoming
diffusive.
In the Delta-Eddington approximation, we
approximate the scattering phase function
by a sum of two parts.
One can derive an alternate diffusion
equation by applying the P1 approximation to
the modified transport equation using the
Delta-Eddington approximation.
V. Venugopalan et al [Phys. Rev. E 58, 2395-2407 (1998)]
have shown that this modified diffusion equation models
scattered light measurements much better than the
standard diffusion equation.
In particular, they showed improvements for small sourcedetector separations thereby extending the spatial range of
validity.
The Fokker-Planck approximation
addresses sharply peaked forward
scattering by asymptotic analysis of the
scattering operator.
Just as the Delta-Eddington approximation
involves two parts, we can add another
operation to obtain the Boltzmann-FokkerPlanck approximation.
See R. Sanchez and N. McCormick [Transport Theory and Statist.
Phys. 12, 129-155 (1983)] for more details about this approximation
and solutions to some inverse problems.
We consider now several computational
approximations that are used for light
propagation in tissues.
Computing numerical solutions to the radiative transport
equation are challenging due to the large number of
independent variables.
There are several numerical methods to solve the radiative
transport equation.
We do not discuss Monte Carlo methods here even though
they are used widely for solving direct problems because
they are not practical for computational solutions of inverse
problems.
There are a variety of methods to treat the
direction and spatial variables numerically.
Directional variables
Spatial variables
• Finite
differences
• Finite volumes
• Finite elements
• Spectral
Combining methods for directional and spatial variables
leads to different numerical methods.
• Spherical
harmonics
• Finite elements
• Discrete ordinate
A comprehensive review of numerical
methods for the radiative transport equation
is not possible here.
Instead, we focus on so-called plane wave solutions and
how they can be used to solve inverse problems.
Plane wave solutions are general solutions to a
homogeneous problem with constant coefficients.
These solutions can be used as a basis for numerical
computations from which we can compute Green’s
function, for example.
Plane wave solutions are special solutions of
the radiative transport equation.
Plane wave solutions have a useful
symmetry.
Plane wave solutions are orthogonal and
complete.
We cannot calculate plane wave solutions
analytically, so we calculate them
numerically using the discrete ordinate
method.
We can compute Green’s function as an
expansion in plane wave solutions.
See A. D. Kim [J. Opt. Soc. Am. A 21, 797-803(2004)] for details of
this calculation.
With Green’s function known, we can make
use of the general representation formula.
Example: Inverse source problem
Under some reasonable assumptions, we can formulate
the optical molecular imaging problem as the inverse
source problem:
Suppose there is a source Q located inside a half-space D
= {z > 0} with boundary  D = {z = 0} of tissue that is
uniformly absorbing and scattering. That is the only
source.
Can we determine properties of the source only from
measurements at the boundary?
We suppose the specific intensity at the
boundary is known (i.e. angular data).
According to the general representation
formula, the intensity at the boundary is
given by
We Fourier transform the reduced problem
and substitute Green’s function to obtain
Using orthogonality of plane wave solutions,
we can construct non-radiating sources.
We compute special solutions of the
reduced problem for an isotropic source.
With the reduced problem, we can compute explicit
solutions for
• Point source
• Pixel/voxel source
Using these two solutions, we seek the location, size and
total strength of a general source.
pixel/voxel source
-function source
Numerical results show the utility of this
method.
We are working currently on extensions of
this idea.
• We are formulating a general inverse scattering theory
for the radiative transport equation using angular data
(with J. Schotland and M. Moscoso).
• We have applied these ideas to reconstructing a “thin”
obstacle in a half space using flux data (with P.
Gonzalez-Rodriguez and M. Moscoso) [AIP 2007 talk].
• We are working with V. Venugopalan (UC Irvine) to
validate this theory using experimental data for layered
media.
Summary
• We have reviewed the theory of radiative transport and
how it is applied to problems in light propagation in
tissues.
• We have discussed some analytical and numerical
approximation methods used to study problems of
practical interest.
• We have discussed plane wave solutions and how they
have use for solving inverse problems.
We recommend four books on the subject
for those getting started in this field.
• K. M. Case and P. F. Zweifel, Linear Transport Theory,
(Addison-Wesley, Reading, MA, 1967).
• A. Ishimaru, Wave Propagation and Scattering in
Random Media, (IEEE Press, New York, 1997).
• A. J. Welch and M. J.C. van Gemert (eds), OpticalThermal Response of Laser-Irradiated Tissue, (Plenum
Press, New York, 1995).
• H. C. van de Hulst, Multiple Light Scattering: Tables,
Formulas, and Applications, (Academic Press, New
York, 1980).