Laminar optical tomography (LOT) is an optical imaging technique capable of making depth-resolved measurements of absorption and fluorescence contrast in scattering tissue. LOT was first demonstrated in 2004 by Hillman et al [1]. The technique combines a non-contact laser scanning geometry, similar to a low magnification confocal microscope, with the imaging principles of diffuse optical tomography (DOT). This thesis describes the development and application of a second generation LOT system, which acquires both fluorescence and multi-wavelength measurements simultaneously and is better suited for in vivo measurements. Chapter 1 begins by reviewing the interactions of light with tissue that form the foundation of optical imaging. A range of related optical imaging techniques and the basic principles of LOT imaging are then described. In Chapter 2, the development of the new LOT imaging system is described including the implementation of a series of interfaces to allow clinical imaging. System performance is then evaluated on a range of imaging phantoms. Chapter 3 describes two in vivo imaging applications explored using the second generation LOT system, first in a clinical setting where skin lesions were imaged, and then in a laboratory setting where LOT imaging was performed on exposed rat cortex. The final chapter provides a brief summary and describes future directions for LOT. LOT has the potential to find applications in medical diagnostics, surgical guidance, and in-situ monitoring owing to its sensitivity to absorption and fluorescence contrast as well as its ability to provide depth sensitive measures. Optical techniques can characterize blood volume and oxygenation, two important biological parameters, through measurements at different wavelengths. Fluorescence measurements, either from autofluorescence or fluorescent dyes, have shown promise for identifying and analyzing lesions in various epithelial tissues including skin [2, 3], colon [4], esophagus [5, 6], oral mucosa [7, 8], and cervix [9]. The desire to capture these types of measurements with LOT motivated much of the work presented here.

This thesis explores two mathematical routes that make the transition from some severely ill-posed parameter reconstruction problems to better-posed versions of them. The general introduction starts by defining what we mean by an inverse problem and its theoretical analysis. We then provide motivations that come from the field of medical imaging. The first part consists in the analysis of an inverse problem involving the Boltzmann transport equation, with applications in Optical Tomography. There we investigate the reconstruction of the spatially-dependent part of the scattering kernel, from knowledge of angularly averaged outgoing traces of transport solutions and isotropic boundary sources. We study this problem in the stationary regime first, then in the time-harmonic regime. In particular we show, using techniques from functional analysis and stationary phase, that this inverse problem is severely ill-posed in the former setting, whereas it is mildly ill-posed in the latter. In this case, we deduce that making the measurements depend on modulation frequency allows to improve the stability of reconstructions. In the second part, we investigate the inverse problem of reconstructing a tensor-valued conductivity (or diffusion) coefficient in a second-order elliptic partial differential equation, from knowledge of internal measurements of power density type. This problem finds applications in the medical imaging modalities of Electrical Impedance Tomography and Optical Tomography, and the fact that one considers power densities is justified in practice by assuming a coupling of this physical model with ultrasound waves, a coupling assumption that is characteristic of so-called hybrid medical imaging methods. Starting from the famous Calderon's problem (i.e. the same parameter reconstruction problem from knowledge of boundary fluxes of solutions), and recalling its lack of injectivity and severe instability, we show how going from Dirichlet-to-Neumann data to considering the power density operator leads to reconstruction of the full conductivity tensor via explicit inversion formulas. Moreover, such reconstruction algorithms only require the loss of either zero or one derivative from the power density functionals to the unknown, depending on what part of the tensor one wants to reconstruct. The inversion formulas are worked out with the help of linear algebra and differential geometry, in particular calculus with the Euclidean connection. The practical pay-off of such theoretical improvements in injectivity and stability is twofold: (i) the lack of injectivity of Calderà³n's problem, no longer existing when using power density measurements, implies that future medical imaging modalities such as hybrid methods may make anisotropic properties of human tissues more accessible; (ii) the improvements in stability for both problems in transport and conductivity may yield practical improvements in the resolution of images of the reconstructed coefficients.

Individuals with schizophrenia exhibit substantial deficits in their ability to perform working memory (WM) tasks, and these deficits have a critical impact on health and life outcomes for these patients, and may be fundamental to the neurophysiological basis of the disorder itself. However, neuroimaging investigations into the nature of these deficits in these patients over the last decade and a half have been stymied by inconsistent findings that leave no clear answer as to their cognitive or neural basis. One hypothesis that has been proposed to account for these inconsistent findings is that the response of some brain regions subserving WM task performance to parametrically increasing WM load, most critically dorsolateral prefrontal cortex, may in fact be non-monotonic in nature; that is, at sufficiently high loads activation in these regions may begin to decrease. If true, this could account for the inconsistent findings in comparisons of patients with schizophrenia and matched controls, as the two groups may be at different points along this putative activation-load 'inverted-U' curve, resulting in different findings depending on the degree of load utilized in any given study. To date, this hypothesis has not been directly tested; however, I report here the results of a series of studies using the self-ordered working memory task that clearly demonstrate such an 'inverted-U' in healthy participants that is absent in patients with schizophrenia. The pattern of findings in the studies reported here are consistent with healthy individuals switching from WM-mediated strategies to long-term memory-mediated strategies as WM load is increased, while patients with schizophrenia fail to make this switch, instead attempting to utilize WM to subserve task performance even when their WM capacity is exceeded.

Peripheral arterial disease (PAD) is a major cause of morbidity and mortality in the USA with an estimated prevalence of up to 20% in those over 75 years. Vascular disease and kidney impairment frequently coexist; prevalence of moderate to severe renal dysfunction in PAD patients is estimated at 27-36%. Knowledge of location, severity, and extent of PAD is imperative for accurate diagnosis and treatment planning. However, all established imaging modalities that are routinely used for treatment planning are contra-indicated in kidney disease patients. Contrast-enhanced x-ray and CT angiography are unsafe due to exposure to nephrotoxic contrast material and ionizing radiation. Recently, the FDA has also warned against the use of gadolinium-enhanced MRA (Gd-MRA) due to evidence that gadolinium could trigger a life-threatening condition known as nephrogenic systemic fibrosis (NSF) in patients with moderate to severe kidney dysfunction. There is a clinical need to develop vascular imaging techniques that are safe in patients with coexisting PAD and renal insufficiency. The focus of this thesis was the development of a non-contrast alternative to Gd-MRA for imaging of peripheral vessels from renal to pedal arteries with MRI. A new imaging sequence for non-contrast visualization of the abdominal and pelvic arteries was designed, implemented, and validated in a small cohort of PAD patients against Gd-MRA. In addition, an existing fast spin-echo based technique for unenhanced imaging of the lower extremities was optimized for improved performance in a clinical setting.

Magnetic resonance spectroscopic imaging (MRSI) is a useful tool for obtaining information on the biochemical processes underlying various pathologies. A widely used multi-voxel localization method is chemical shift imaging (CSI) which uses gradients for phase encoding. Although simple to implement, low in specific absorption rate (SAR) and immune to chemical shift displacement (CSD), it also suffers from some well known drawbacks caused by its sinc-shaped point spread function (PSF). This results in loss of both signal-to-noise ratio (SNR) as well as localization, an effect that is exacerbated at low resolutions. In contrast, an alternative localization method, Hadamard spectroscopic imaging (HSI) benefits from a theoretically ideal PSF and consequently does not suffer from these drawbacks. In this work we exploit the theoretically ideal PSF of HSI encoding to develop a novel three dimensional (3D) multi-voxel MR localization method based on transverse HSI (T-HSI). The advantages of T HSI are that unlike gradient phase-encoding: (i) the volume of interest (VOI) does not need to be smaller than the field-of-view to prevent aliasing; (ii) the number of partitions in each direction can be small, 8, 4 or even 2 at no cost in PSF; (iii) the VOI does not have to be contiguous; and (iv) the voxel profile depends on the available B1 and pulse synthesis paradigm and can therefore, at least theoretically, approach "ideal" "1" inside and "0" elsewhere. Clinical utility of the new method is shown by spectra obtained from the brain of a healthy volunteer. The benefits of T-HSI are demonstrated by a quantitative comparison to CSI of the SNR and localization in a phantom in both one and three dimensions at clinical resolutions. A novel matrix formalism is used to quantify the impact of non-ideal flip angles on T-HSI. The superior PSF of T-HSI is then used to demonstrate the feasibility of scanning regions near or on the skull while limiting the impact of lipid contamination and obtaining quantifiable spectra. A comparison to spectra obtained using CSI is shown for a healthy volunteer. The new method is also used in a clinical pathology: to scan multiple sclerosis (MS) lesions occurring near the skull. To maintain the benefits provided by the PSF of HSI at higher fields, despite its susceptibility to CSD, a additional hybrid sequence is also developed that limits both the SAR and the CSD, regardless of the size of the VOI. A comparison to CSI in a phantom and in-vivo is carried out and spectra obtained from the brain of a healthy volunteer at 3T are shown. Finally, future research avenues involving extension of this research to ultra high fields (7T) are discussed and possible clinical uses are described.

Diffuse optical tomography (DOT) is an imaging technique where near infrared (NIR) photons are used to probe biological tissue. DOT allows for the recovery of three-dimensional maps of tissue optical properties, such as tissue absorption and scattering coefficients. The application of DOT as a tool to aid in the diagnosis of rheumatoid arthritis (RA) is explored in this work. Algorithms for improving the image reconstruction process and for enhancing the clinical value of DOT images are presented in detail. The clinical data considered in this work consists of 99 fingers from subjects with RA and 120 fingers from healthy subjects. DOT scans of the proximal interphalangeal (PIP) joint of each finger is performed with modulation frequencies of 0, 300, and 600 MHz. A computer-aided diagnosis (CAD) framework for extracting heuristic features from DOT images and a method for using these same features to classify each joint as affected or not affected by RA is presented. The framework is applied to the clinical data and results are discussed in detail. Then, an algorithm for recovering the optical properties of biological media using the simplified spherical harmonics (SPN) light propagation model is presented. The computational performance of the algorithm is analyzed and reported. Finally, the SPN reconstruction algorithm is applied to clinical data of subjects with RA and the resulting images are analyzed with the CAD framework. As the first part of the CAD framework, heuristic image features are extracted from the absorption and the scattering coefficient images using multiple compression and dimensionality reduction techniques. Overall, 594 features are extracted from the images of each joint. Then, machine-learning techniques are used to evaluate the ability to discriminate between images of joints with RA and images of healthy joints. An evolution-strategy optimization algorithm is developed to evaluate the classification strength of each feature and to find the multidimensional feature combination that results in optimal classification accuracy. Classification is performed with k-nearest neighbors (KNN), linear (LDA) and quadratic discriminate analysis (QDA), self-organizing maps (SOM), or support vector machines (SVM). Classification accuracy is evaluated based on diagnostic sensitivity and specificity values. Strong evidence is presented that suggest there are clear differences between the tissue optical parameters of joints with RA and joints without RA. It is first shown that data obtained at 600 MHz leads to better classification results than data obtained at 300 and 0 MHz. Analysis of each extracted feature shows that DOT images of subjects with RA are statistically different (p < 0.05) from images of subjects without RA for over 90% of the features. Evidence shows that subjects with RA that do not have detectable signs of erosion, effusion, or synovitis (i.e. asymptomatic subjects) in MRI and US images have optical profiles similar to subjects who do have signs of erosion, effusion, or synovitis; furthermore, both of these cohorts differ from healthy controls subjects. This shows that it may be possible to accurately identify asymptomatic subjects with DOT scans. In contrast, these subjects remain difficult to identify from MRI and US images. The implications of these results are profound, as they suggest it may be possible to identify RA with DOT at an earlier stage compared to standard imaging techniques. Results from the feature-selection algorithm show that the SVM algorithm (with a third order polynomial kernel) achieves 100.0% sensitivity and 97.8% specificity. Lower bounds for these results (at 95.0% confidence level) are 96.4% and 93.8%, respectively. Image features most predictive of RA are from the spatial variation of optical properties and the absolute range in feature values. The optimal classifiers are low dimensional combinations (< 7 features). Robust cross- validation is performed to ensure the generalization of these classification results. The SPN -based reconstruction algorithm uses a reduced-Hessian sequential quadratic programming (rSQP) PDE-constrained optimization approach to maximize computational efficiency. The complex-valued forward model, or frequency domain SPN equations (N = 1, 3), is discretized using the finite-volume method and solved on unstructured computational grids using the restarted GMRES algorithm. The image reconstruction algorithm is presented in detail and its performance benchmarked against the ERT algorithm. The algorithm is subsequently used to recover the absorption and scattering coefficient images of joints scanned in the RA clinical study. While the SPN model is inherently less accurate than the ERT model, it is nevertheless shown that the images obtained with the SP3-based reconstruction algorithm are sufficiently accurate and allow for the diagnosis of RA at clinically relevant sensitivity [87.9% (78.1%, 100.0%)] and specificity [92.9% (84.6%, 100.0%)] values (the 95.0% confidence interval is specified in brackets). In contrast to results obtained with the SP3 model, the images generated with the SP1 algorithm yield significantly lower sensitivity [66.7% (46.6%, 100.0%)] and specificity [81.0% (64.8%, 100.0%)] values. While some numerical accuracy is sacrificed by selecting the SP3 model over the ERT model, the superior computational performance of the SP3 algorithm allows for computation of the absorption and the scattering coefficient images in under 15 minutes and requires less than 200 MB of RAM per finger (compared to the over 180 minutes and over 6 GB of RAM needed by the ERT-based algorithm). Overall, results indicate that the SP3-based reconstruction algorithm provides computational advantages over the ERT-based algorithm without sacrificing significant classification accuracy. In contrast, the SP1 model provides computational advantages compared to the ERT at the expense of classification accuracy. This indicates that the frequency-domain SP3 model is an ideal light propagation model for use in DOT scanning of finger joints with RA. Altogether, the results presented in this dissertation underscore the high potential for DOT to become a clinically useful diagnostic tool. The algorithms and framework developed as part of this dissertation can be directly used on future data to help further validate the hypotheses presented in this work and to further establish DOT imaging as a valuable diagnostic tool.

It is well known that the radiative transfer equation (RTE) is the most accurate deterministic light propagation model employed in diffuse optical tomography (DOT). RTE-based algorithms provide more accurate tomographic results than codes that rely on the diffusion equation (DE), which is an approximation to the RTE, in scattering dominant media. However, RTE based DOT (RTE-DOT) has limited utility in practice due to its high computational cost and lack of support for general non-contact imaging systems. In this dissertation, I developed fast reconstruction algorithms for RTE-based DOT (RTE-DOT), which consists of three independent components: an efficient linear solver for forward problems, an improved optimization solver for inverse problem, and the first light propagation model in free space that fully considers the angular dependency, which can provide a suitable measurement operator for RTE-DOT. This algorithm is validated and evaluated with numerical experiments and clinical data. According to these studies, the novel reconstruction algorithm is up to 30 times faster than traditional reconstruction techniques, while achieving comparable reconstruction accuracy.

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Intensity Modulated Radiation Therapy (IMRT) is a complex form of radiation delivery for the treatment of malignant tumours and other diseases. In IMRT treatment planning, quantitative assessment is crucial to measure and improve the plan quality and treatment delivery. The search for simple and universal quantitative metrics to assess IMRT treatment plan quality has been identified as important but as yet not entirely successful. The aim of this thesis was to assess the IMRT treatment plan quality by establishing quantitative metrics for planning conformity and complexity. The metrics proposed in this work were simple, reproducible and universally applicable to all IMRT techniques, which included step-and-shoot IMRT (SSIMRT), volumetric modulated arc therapy (VMAT) and helical tomotherapy (HT). Two metrics, conformity index (CI) and conformation number (CN) were adopted to quantify the plan conformity. The data used for CI and CN calculations were easily retrieved from dose volume histogram (DVH). By reporting both of these metrics, comprehensive information on target coverage and irradiation of normal tissues could be provided. For the quantification of planning complexity, a new and novel spatial complexity matrix (SCM) was introduced to measure the average dose gradient of a dose profile. In addition, the spatial frequency ratio (SFR) was established to explore the proportion of rapidly varying dose with distance in a treatment plan by using one-dimensional power spectral density (1D PSD). Virtual phantoms were developed for the initial quantitative assessment, in order to form a basis for treatment plan inter-comparisons amongst the different IMRT techniques. A series of multi organs at risk (OARs) phantoms was developed to simulate the planning target volume (PTV) and OARs for different configurations. A virtual prostate phantom was also designed to include a unique shape of PTV and the OAR in close proximity to PTV, in order to mimic clinical prostate case. Quantitative assessments were undertaken on all the IMRT plans generated using the virtual phantoms. The results of these phantom studies have shown for the first time, the feasibility of the developed quantitative metrics for assessing plan quality. Following the successful application of SCM and SFR on the phantom plans, verification work was undertaken to demonstrate the clinical relevance of these self-developed complexity metrics. A retrospective study was carried out to assess the complexity of plans for the treatment of prostate and head and neck tumours. The information contained in DICOM-RT objects were utilised to acquire dose data from the corresponding dose plane. A qualitative survey on plan complexity was also conducted amongst treatment planners, to demonstrate the correlation between the qualitative and quantitative results. These preliminary studies demonstrated the successful application of the self-developed complexity metrics on clinical IMRT treatment plans. In conclusion, the work in this thesis has demonstrated the successful establishment of quantitative metrics for assessing plan conformity and complexity of different IMRT techniques. These metrics were considered as universal tools for the inter-comparison of plan quality for different IMRT techniques and were successfully applied and translated from phantom studies to the clinical setting. Whilst the judgment and experience of the treatment planner undoubtedly remains paramount for making a final decision on the best plan in the interest of the patient, it is expected that the use of quantitative metrics will provide an effective means of benchmarking performance, minimising treatment plan variability and enhancing the quality of IMRT treatment planning.