| Summary: | Optical Tomography is an imaging modality that enhances early diagnosis of disease through use of harmless Near-Infrared rays instead of conventional x-rays. The subsequent images are used to reconstruct the object. However Optical Tomography has not been effectively utilized due to the complicated photon scattering phenomenon and ill-posed nature of the corresponding image reconstruction scheme.
The major method for reconstruction of the object is based on an iterative loop that constantly minimizes the difference between the predicted model of photon scattering with acquired images. Currently the most effective method of predicting the photon scattering pattern is the solution of the Radiative Transfer Equation (RTE) using the Finite Elements Method (FEM). However, the conventional FEM uses classical C0 interpolation functions, which have shortcomings in terms of continuity of the solution over the domain as well as proper representation of geometry. Hence higher discretization is necessary to maintain accuracy of gradient-based results which may significantly increase the computational cost in each iteration.
This research implements the recently developed Isogeometric Approach (IGA) and particularly IGA-based FEM to address the aforementioned issues. The IGA-based FEM has the potential to enhance adaptivity and reduce the computational cost of discretization schemes. The research in this study applies the IGA method to solve the RTE with the diffusion approximation and studies its behavior in comparison to conventional FEM.
The results show comparison of the IGA-based solution with analytical and conventional FEM solutions in terms of accuracy and efficiency. While both methods show high levels of accuracy in reference to the analytical solution, the IGA results clearly excel in accuracy. Furthermore, FE solutions tend to have shorter runtimes in low accuracy results. However, in higher accuracy solutions, where it matters the most, the IGA proves to be considerably faster. === Doctor of Philosophy === CT scans can save lives by allowing medical practitioners observe inside the patient's body without use of invasive surgery. However, they use high energy, potentially harmful x-rays to penetrate the organs. Due to limits of the mathematical algorithm used to reconstruct the 3D figure of the organs from the 2D x-ray images, many such images are required. Thus, a high level of x-ray exposure is necessary, which in periodic use can be harmful.
Optical Tomography is a promising alternative which replaces x-rays with harmless Near-infrared (NIR) visible light. However, NIR photons have lower energy and tend to scatter before leaving the organs. Therefore, an additional algorithm is required to predict the distribution of light photons inside the body and their resulting 2D images. This is called the forward problem of Optical Tomography. Only then, like conventional CT scans, can another algorithm, called the inverse solution, reconstruct the 3D image by diminishing the difference between the predicted and registered images.
Currently Optical Tomography cannot replace x-ray CT scans for most cases, due to shortcomings in the forward and inverse algorithms to handle real life usages. One obstacle stems from the fact that the forward problem must be solved numerous times for the inverse solution to reach the correct visualization. However, the current numerical method, Finite Element Method (FEM), has limitations in generating accurate solutions fast enough using economically viable computers. This limitation is mostly caused by the FEM's use of a simpler mathematical construct that requires more computations and is limited in accurately modelling the geometry and shape.
This research implements the recently developed Isogeometric Analysis (IGA) and particularly IGA-based FEM to address this issue. The IGA-based FEM uses the same mathematical construct that is used to visualize the geometry for complicated applications such as some animations and computer games. They are also less complicated to apply due to much lower need for partitioning the domain. This study applies the IGA method to solve the forward problem of diffuse Optical Tomography and compare the accuracy and speed of IGA solution to the conventional FEM solution. The comparison reveals that while both methods can reach high accuracy, the IGA solutions are relatively more accurate. Also, while low accuracy FEM solutions have shorter runtimes, in solutions with required higher accuracy levels, the IGA proves to be considerably faster.
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