There is still a great need for developing suitable data processing algorithms in optical tomography. Prof. Hielscher's group has pioneered so-called model-based iterative image (MOBIIR) reconstructions methods. Iterative reconstruction schemes consist usually of three components. The first component is a forward model that provides a prediction of the measurements based on a guess of the system parameters (here the spatial distributions of ma and ms).  Second, a scheme is used that compares the predicted data with the measured data, which results in some sort of error function, often also called objective function or norm.  The third component is an efficient way of updating the system parameters of the forward model, which in turn provides a new set of predicted measurement data. The Biophotonics and Optical Radiology Laboratory focuses on the use of the equation of radiative transfer (ERT) as a forward model of light propagation. Working with this equation is extremely challenging and most researchers employ the diffusion approximation, which leads to simpler algorithms that converge much faster. However, the diffusion approximation fails to accurately describe light propagation in media that contain void-like regions with low scattering, or high absorbing regions. Practical examples where diffusion theory fails include brain imaging, imaging of joint diseases, and the fast growing field of small animal imaging.  By developing ERT based theoretical models, the BORAL group is capable of providing optical image reconstruction algorithms for these important applications. There are two major algorithms in place that are frequently used in all clinical and small animal studies at Columbia. First, there is a finite-differencing code that employs a method of reverse differentiation, and a second code is based on finite-element methods and is particularly suited for objects with arbitrary geometries. Beyond these two major existing codes Prof. Hielscher's team is currently developing three new algorithms. First, there is a so-called frequency-domain transport code. In this case the source is amplitude modulated in the range of 100 –1000 Mhz, which leads to the propagation of photon-density waves in tissues. Many instruments on the market and in research laboratories make use of this approach. By measuring the phase of the waves in addition to the amplitude, these instruments provide more information than so called steady-state technologies which don’t modulate the light source. Only diffusion theory based codes exist so far, which prevents the application of this approach to the cases outlined above. A second area of development is molecular fluorescence  imaging, which promises to bring about a revolution in medical imaging. In fluorescence molecular tomography, light from an external source is absorbed by a fluorophore and reemitted at a longer wavelength. The emitted light is detected by sensors on the surface of the tissue. This data is then used to find the distribution of light sources (generated by fluorescence markers) inside the tissues. The BORAL team has published first papers on the theoretical aspects of this problem. Finally, Prof. Hielscher's group is currently exploring image reconstruction approaches that go beyond the MOBIIR approach. In a recent paper they reported on the implementation of an augmented Lagrangian approach for solving the image reconstruction problem in OT. In this case the inverse problem is formulated as a minimization problem with the ERT being considered as an equality constraint on the set of  “optical properties – radiance” pairs. Compared to the traditional approaches for optical tomographic imaging where one solves several forward and adjoint problems at each optimization iteration, the PDE-constrained method solves the forward and inverse problems simultaneously. In initial simulation, a significant acceleration of the reconstruction process, compared to our previously reported codes, was achieved.

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