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Optical Tomography
What is Optical Tomography
Optical tomography, is centred around the simple idea that light passes through the body in small amounts, and emerges bearing clues about tissues through which it has passed. It involves the technique of using near infrared photons for imaging specific parts of the body to obtain information about tissue abnormalities, such as breast and brain tumours. Optical tomographic imaging is performed by obtaining a spatially resolved map of the optical properties within the part of the body under investigation.
Why Optical Tomography
Medical images of living tissue can be produced using any of the existing techniques such as CT (computed x-ray tomography), PET (Positron Emission Tomography), Ultrasound or MRI (Magnetic Resonance Imaging). All these techniques are powerful and have great advantages, but they suffer from drawbacks which limit their use as continuous, non-invasive, portable and low cost medical monitors.
Light is nonionizing over many wavelengths and is known to function well as a medical probe. In particular, red and infrared light pass relatively easily through such structures as the skull, brain and breast, and is well tolerated in large doses. It can, therefore, be used to quantify tissue properties. Near-infrared optical tomography has been recognised as an ideal non-invasive diagnostic technique due to its potential low-cost and very little side-effects
How does Optical Tomography work
Getting near-infrared optical tomography to work, is a big
challenge, for the associated mathematical problem is by its
nature very difficult to model accurately. Imaging in x-ray,
is a relatively straight forward matrix inversion, since the
radiation is assumed to have travelled along the straight line
between the source and detector. In optical tomography, information
retrieved from highly scattered light poses a formidable problem.
Biological tissue is a highly scattering medium, causing the
photons to take lengthy and highly irregular paths; the optical
pathlength is usually several times the geometric distance between
the input and output.
Figure 1 shows an ultrashort laser pulse propagation
through scattering medium spreading into ballistic photons (travel
along the straight line path and arrive first) and snake
photons (travel zig zag paths slightly off the straight
line path and arrive second), and diffuse photons (undergo
random walks and arrive last)
Multiple scattering events occur for all photons propagating through tissue, this produce a wide range in paths taken by, and time required for, different photons to traverse tissue. Since photon path lengths vary with absorbency and scattering, it is difficult to predict such paths. In addition to this the fact that the scattering medium through which light passes includes low-scattering (CSF) regions, then development of rigorous light transport models is the only way forward.
The imaging system
It is clear from Figure 1 that an optical tomographic system
requires a source generation device, an interface to deliver
the light to a small point of tissue surface, another interface
to pick up the signal at another surface point, and a detector
to quantify the response signal. The light source will be a NIR
laser capable of producing either ultrafast laser pulses (time
domain technique) as shown in Figure 1, or an intensity modulated
signal (frequency domain signal). Fibre-optics are used for the
transfer of light from the laser source to the tissue and from
tissue to the detector. The detector must be highly time-resolving
for a time-domain system (usually a streak camera), or a device
to compare phase shifts for frequency domain systems.
Optical Tomography is done by following two main steps. The
forward problem consists of detailed description of the sample
under investigation; and includes its experimental geometry,
source and detector characteristics and its optical properties
which determine the measurement results. The second step is the
Inverse Problem; this is done by taking a set of measurements
and error estimates, limited parameters describing the sample
and experiment (geometry, source/detector configuration), and
deriving the remaining parameters (the internal optical properties).
This is illustrated in Figure 2.
Different Modelling Techniques
The interaction of light and tissue is described fundamentally by Maxwell's equations of the electromagnetic field in a continuos medium with a random dielectric constant. Direct solutions from Maxwell's equations have hardly been attempted. Instead several differential or integro-differential equations have been used among them the radiation transport equation and the diffusion approximation which is derived from the transport equation. The use of the diffusion approximation has led to considerable understanding of photon migration in tissue and to significant advances in optical tomography. It has been shown, however, to be inaccurate in many instances, particularly for applications that involve structures near the surface, or situations with fluids and vessels and at any sharp discontinuity of scattering property. Another technique for modelling the interaction of light and tissue is based on the Monte Carlo (MC) methods, which treat light as a collection of discrete particles. Such methods have been traditionally considered more accurate, physically realistic and has been used by a large number of investigators in the field. It involves extensive computing time since a large number of photons must be followed through the media, hence the computer time required increases exponentially with the size of the sample being modelled.
What is new about our work
A higher-order photon transport method
In this study, the full higher-order radiation transport solution as produced by the spherical harmonics (PN) method is used to model the photon propagation problem. This is an elegant and well established method for modelling particle's angular dependence. It's use has already been considered by other workers in the field, but only with limited success. This is largely due to the choice of a numerically inefficient framework. The PN method employed here, is based on a variational formulation for the second-order form of the Boltzmann transport equation, which is the most natural and numerically efficient way of deriving higher-order approximations. It's generality enables one to go to very high orders of angular approximation, for modelling not only particle angular dependence but also complex scattering phase functions. The method is combined with a finite element representation in space, which gives it geometrical flexibility matched by Monte Carlo methods only. Although these provide equally rigorous treatment of photon migration they are several orders of magnitude slower and produce limited amount of information.
A computer program based on the FE-PN (Finite element-spherical harmonics method), EVENT, has been developed for the solution of steady-state and time dependent direct and /or adjoin radiation transport problems in different geometry: (1D) slab, spherical, infinite cylindrical; (2D) x-y, r-z; and (3D) x-y-z. Isoparametric finite elements are used to represent the spatial dependence of the angular flux. The elements available consist of lines, triangles, quadrilaterals, tetrahedral, hexahedron and prisms. The program provides several economical sparse-matrix serial and parallel methods based on gaussian elimination or preconditioned conjugate gradient procedures for the solution of linear system of equations, a multi-frequency capability is also provided. The FENTRRE (Finite ElemeNt TRansport theory REconstruction) is used in the reconstruction shown in Figures 4 and 5.
Some results from our work
Comparison between the forward results obtained with Monte
Carlo (MC), FE-PN and diffusion approximation for
a 3D phantom is presented in Figure 3.
Figure 3 : (Left) Shows a phantom (Delrin), consists of a homogeneous cylinder 4 mm high and 10 mm radius. At t = 0 an isotropic source of radius 0.4 mm is pulsed for 1 ps at the centre of one of the surfaces. The time evolution of the exiting current at the corresponding opposite end is followed. (Right) Showing the results from Monte Carlo (MC), diffusion (Diff.), and P5 approximations. It can be noticed that the diffusion theory is not valid in this case because of the small thickness of the phantom.
Figure 4 shows A 2-D reconstruction of the optical properties utilising time-dependent data for 16 sources and 16 detectors arrangements, using P7 angular approximation.
Figure 4 : (Left) Infinite cylinder of 27mm radius with 3 cylindrical regions of 2.5mm radius pulsed with 1ps beams. The scattering and absorption cross sections are (in mm-1): for cylinder 3.12 and 0.02; 1st region 2.9 and 0.2; 2nd region 2.9 and 0.3; 3rd region 2.9 and 0.1. The mean cosine, g is 0.92. (Right:) Reconstruction of optical properties from the Monte Carlo time-dependent data for a 16 sources and 16 detectors arrangement, using P7 angular approximation and 32x32 pixel mesh..
Figure 5 :(Left) 27mm radius infinite cylinder with 1 cylindrical region of 2.5mm radius is pulsed with 1ps beam. The scattering and absorption cross sections are (in mm-1): cylinder 3.12 and 0.02; cylindrical region 3.12 and 0.2; The mean cosine, g is 0.5. (Right) Reconstruction of optical properties from the Monte Carlo time-dependent data for a 16 sources and 16 detectors arrangement, using P7 angular approximation and 24x24 pixel mesh.. A uniform mesh consisting of 4751 nodes and 9308 linear triangular elements was used. Reconstruction took approximately 10 cpu minutes for 35 iterations.
Our survey has shown that the reconstructed images published in the literature so far are limited to the localisation of inhomogeneities rather than an absolute quantification of their optical properties; we are hopeful that FENTRRE used in this study is capable of producing a better representation of the quantitative values in addition to the localisation.
This page has been designed by Khadija Tahir Applied Optics Group, Blackett Laboratory Imperial College, London SW7 2BZ