Optical tomography reconstruction algorithm based on the radiative transfer equation considering refractive index—Part 1: Forward model
Introduction
Optical tomography (OT) is a medical imaging modality that calculates three-dimensional (3D) maps of absorption and scattering coefficients in biological tissue by using a radiative transfer model for visible or near-infrared light [1], [2]. The mathematical framework of OT contain a forward model for light propagation and an inverse model for reconstructing the optical property map from predicted and measured partial boundary currents [1]. The most commonly applied forward model in OT has been the diffusion equation due to its mathematical simplicity and the availability of a vast amount of fast and efficient numerical solver. The diffusion equation is a low-order approximation to the more generally applicable radiative transfer equation (RTE), so it is only valid in the diffusion limit wherein scattering dominates absorption. In the most general terms, the light propagation in biological tissue can be described by the radiative transfer equation [3].
The RTE is typically solved with some numerical methods since no analytical methods of RTE exist for non-uniform medium with complex geometries [1], [2], [3], [4]. These numerical methods are either low-order approximation to the RTE, such as the diffusion approximation, or high-order approximation to the RTE, e.g. discrete ordinates (SN) methods. The spatial discretization of the tissue domain can be performed with either finite-difference (FD), finite-volume (FV), or finite-element (FE) techniques.
Recently, the research for OT is only done on the absorption and scattering coefficients. However, not only the absorption and scattering coefficients, but also the refractive index have effect on the optical property for optical tomography reconstruction. Depend on the physical properties of biological tissue or temperature, the refractive index may be a function of spatial location [4], [5], [6], [7], [8], [9].
In our present study, we selected the high-order approximation to the radiative transfer equation, that is the discrete ordinates (SN) methods. We used the upwind-difference method to formulate the RTE considering the refractive index. The upwind-difference method is the high order approximation of the radiative transfer equation, it has the advantage that it provides more convenient mathematical framework for calculating the derivative of the fluence with respect to the optical parameters using an adjoint differentiation technique [10], [11], [12], [13], [14]. What's more, the upwind-difference method is a backward difference method, it has another advantage that it is diagonal dominance which can avoid appearing oscillation or convergence in the process of discretization [15], [16], [17], [18], [19].
Section snippets
Uniform refractive index model
A beam of light of intensity I of a given wavelength λ travels along a given direction at the speed of light c/n in the medium (n is the refractive index). Along its path, a fraction is absorbed (μaI is the amount of light intensity absorbed in a unit distance). Another fraction is scattered (μsI), i.e. redirected in another direction after interaction with the medium. Only consistent diffusion is considered, meaning that a scattering event affecting a beam does not impact its wavelength. This
Numerical experiments
In order to verify the accuracy of the forward algorithm, we selected a human skull phantom in Fig. 1(a), the image contain three human skulls, where the relatively large bright oval is the external contours of the skull, the relatively small black oval is the tumors, the lighter circle is of a void-like region in human brain tissue. First we calculate the projection of Fig. 1(a), then we obtain Fig. 1(b).
Results
In this paper, we have performed some preliminary simulations using MATLAB for both cases of the uniform refractive index and gradient refractive index. In these two cases, we assume the domain Ω ⊂ R2 to be divided into the I × J = 256 × 256 grid regions. We simulated a human brain phantom image that contains void-like region and tumor under the conditions that the scattering coefficient μs = 58 cm−, the absorption coefficient μa = 0.35cm−1, the anisotropy coefficient g = 0.8, the iteration step along the x
Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 11271141) and the Natural Science Foundation of Guangdong Province, China (No. 9151064201000040).
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Optical tomography reconstruction algorithm based on the radiative transfer equation considering refractive index: Part 2. Inverse model
2013, Computerized Medical Imaging and GraphicsCitation Excerpt :Using the adjoint difference method, the gradient is calculated by a simple scalar product, and it can avoid computing the sensitivities. When it is need to consider the sensitivities, what we should do is deduce the expressions of the gradient [15]. In the following, we calculate the gradient in two cases, that is the case of uniform refractive index and the case of gradient refractive index.