A principal component approach to space-based gravitational wave astronomy

The current approach to data analysis for the Laser Interferometry Space Antenna (LISA) depends on the time delay interferometry observables (TDI) which have to be generated before any weak signal detection can be performed. These are linear combinations of the raw data with appropriate time shifts that lead to the cancellation of the laser frequency noises. This is possible because of the multiple occurrences of the same noises in the different raw data. Originally, these observables were manually generated starting with LISA as a simple stationary array and then adjusted to incorporate the antenna's motions. However, none of the observables survived the flexing of the arms in that they did not lead to cancellation with the same structure. The principal component approach is another way of handling these noises that was presented by Romano and Woan which simplified the data analysis by removing the need to create them before the analysis. This method also depends on the multiple occurrences of the same noises but, instead of using them for cancellation, it takes advantage of the correlations that they produce between the different readings. These correlations can be expressed in a noise (data) covariance matrix which occurs in the Bayesian likelihood function when the noises are assumed be Gaussian. Romano and Woan showed that performing an eigendecomposition of this matrix produced two distinct sets of eigenvalues that can be distinguished by the absence of laser frequency noise from one set. The transformation of the raw data using the corresponding eigenvectors also produced data that was free from the laser frequency noises. This result led to the idea that the principal components may actually be time delay interferometry observables since they produced the same outcome, that is, data that are free from laser frequency noise. The aims here were (i) to investigate the connection between the principal components and these observables, (ii) to prove that the data analysis using them is equivalent to that using the traditional observables and (ii) to determine how this method adapts to real LISA especially the flexing of the antenna. For testing the connection between the principal components and the TDI observables a 10x 10 covariance matrix containing integer values was used in order to obtain an algebraic solution for the eigendecomposition. The matrix was generated using fixed unequal arm lengths and stationary noises with equal variances for each noise type. Results confirm that all four Sagnac observables can be generated from the eigenvectors of the principal components. The observables obtained from this method however, are tied to the length of the data and are not general expressions like the traditional observables, for example, the Sagnac observables for two different time stamps were generated from different sets of eigenvectors. It was also possible to generate the frequency domain optimal AET observables from the principal components obtained from the power spectral density matrix. These results indicate that this method is another way of producing the observables therefore analysis using principal components should give the same results as that using the traditional observables. This was proven by fact that the same relative likelihoods (within 0.3%) were obtained from the Bayesian estimates of the signal amplitude of a simple sinusoidal gravitational wave using the principal components and the optimal AET observables. This method fails if the eigenvalues that are free from laser frequency noises are not generated. These are obtained from the covariance matrix and the properties of LISA that are required for its computation are the phase-locking, arm lengths and noise variances. Preliminary results of the effects of these properties on the principal components indicate that only the absence of phase-locking prevented their production. The flexing of the antenna results in time varying arm lengths which will appear in the covariance matrix and, from our toy model investigations, this did not prevent the occurrence of the principal components. The difficulty with flexing, and also non-stationary noises, is that the Toeplitz structure of the matrix will be destroyed which will affect any computation methods that take advantage of this structure. In terms of separating the two sets of data for the analysis, this was not necessary because the laser frequency noises are very large compared to the photodetector noises which resulted in a significant reduction in the data containing them after the matrix inversion. In the frequency domain the power spectral density matrices were block diagonals which simplified the computation of the eigenvalues by allowing them to be done separately for each block. The results in general showed a lack of principal components in the absence of phase-locking except for the zero bin. The major difference with the power spectral density matrix is that the time varying arm lengths and non-stationarity do not show up because of the summation in the Fourier transform.

Eulerian-Lagrangian definition of coarse bed-load transport: Theory and verification with low-cost inertial measurement units

Fluvial sediment transport is controlled by hydraulics, sediment properties and arrangement, and flow history across a range of time scales. This physical complexity has led to ambiguous definition of the reference frame (Lagrangian or Eulerian) in which sediment transport is analysed. A general Eulerian-Lagrangian approach accounts for inertial characteristics of particles in a Lagrangian (particle fixed) frame, and for the hydrodynamics in an independent Eulerian frame. The necessary Eulerian-Lagrangian transformations are simplified under the assumption of an ideal Inertial Measurement Unit (IMU), rigidly attached at the centre of the mass of a sediment particle. Real, commercially available IMU sensors can provide high frequency data on accelerations and angular velocities (hence forces and energy) experienced by grains during entrainment and motion, if adequately customized. IMUs are subjected to significant error accu- mulation but they can be used for statistical parametrisation of an Eulerian-Lagrangian model, for coarse sediment particles and over the temporal scale of individual entrainment events. In this thesis an Eulerian-Lagrangian model is introduced and evaluated experimentally. Absolute inertial accelerations were recorded at a 4 Hz frequency from a spherical instrumented particle (111 mm diameter and 2383 kg/m3 density) in a series of entrainment threshold experiments on a fixed idealised bed. The grain-top inertial acceleration entrainment threshold was approximated at 44 and 51 mg for slopes 0.026 and 0.037 respectively. The saddle inertial acceleration entrainment threshold was at 32 and 25 mg for slopes 0.044 and 0.057 respectively. For the evaluation of the complete Eulerian-Lagrangian model two prototype sensors are presented: an idealised (spherical) with a diameter of 90 mm and an ellipsoidal with axes 100, 70 and 30 mm. Both are instrumented with a complete IMU, capable of sampling 3D inertial accelerations and 3D angular velocities at 50 Hz. After signal analysis, the results can be used to parametrize sediment movement but they do not contain positional information. The two sensors (spherical and ellipsoidal) were tested in a series of entrainment experiments, similar to the evaluation of the 111 mm prototype, for a slope of 0.02. The spherical sensor entrained at discharges of 24.8 ± 1.8 l/s while the same threshold for the ellipsoidal sensor was 45.2 ± 2.2 l/s. Kinetic energy calculations were used to quantify the particle-bed energy exchange under fluvial (discharge at 30 l/s) and non-fluvial conditions. All the experiments suggest that the effect of the inertial characteristics of coarse sediments on their motion is comparable to the effect hydrodynamic forces. The coupling of IMU sensors with advanced telemetric systems can lead to the tracking of Lagrangian particle trajectories, at a frequency and accuracy that will permit the testing of diffusion/dispersion models across the range of particle diameters.

An investigation of the relationship between the components of tumour microenvironment, signal transduction pathways and outcome in breast cancer

Breast cancer, the most commonly diagnosed type of cancer in women, is a major cause of morbidity and mortality in the western world. Well-established risk factors of breast cancer are mostly related to women’s reproductive history, such as early menarche, late first pregnancy and late menopause. Survival rates have improved due to a combination of factors, including better health education, early detection with large-scale use of screening mammogram, improved surgical techniques, as well as widespread use of adjuvant therapy. At initial presentation, clinicopathological features of breast cancer such as age, nodal status, tumour size, tumour grade, and hormonal receptor status are considered to be the standard prognostic and predictive markers of patient survival, and are used to guide appropriate treatment strategies. Lymphovascular invasion (LBVI), including lymphatic (LVI) and blood (BVI) vessel invasion, has been reported to be prognostic and merit accurate evaluation, particularly in patients with node negative tumours who might benefit from adjuvant chemotherapy. There is a lack of standard assessment and agreement on distinguishing LVI from BVI despite the major challenges in the field. A systematic review of the literatures, examining methods of detection and the prognostic significance of LBVI, LVI and BVI, was carried out. The majority of studies used haematoxylin and eosin (H&E) and classical histochemistry to identify LVI and BVI. Only few recent studies used immunohistochemistry (IHC) staining of the endothelium lining lymphatic and blood vessels, and were able to show clear differences between LVI and BVI. The prognostic significance of LBVI and LVI was well-documented and strongly associated with aggressive features of breast tumours, while the prognostic value and the optimal detection method of BVI were unclear. Assessment and prognostic value of LBVI on H&E sections (LBVIH&E) was examined and compared to that of LVI and BVI detected using IHC with D2-40 for LVI (LVID2–40) and Factor VIII for BVI (BVIFVIII) in patients with breast cancer including node negative and triple negative patients (n=360). LBVIH&E, LVID2–40 and BVIFVIII were present in 102 (28%), 127 (35%) and 59 (16%) patients respectively. In node negative patients (206), LBVIH&E, LVID2–40 and BVIFVIII were present in 41 (20%), 53 (26%) and 21 (10%) respectively. In triple negative patients (102), LBVIH&E, LVID2–40 and BVIFVIII were present in 35 (29%), 36 (35%) and 14 (14%) respectively. LBVIH&E, LVID2–40 and BVIFVIII were all significantly associated with tumour recurrence in all cohorts. On multivariate survival analysis, only LVID2–40 and BVIFVIII were independent predictors of cancer specific survival (CSS) in the whole cohort (P=0.022 and P<0.001 respectively), node negative (P=0.008 and P=0.001 respectively) and triple negative patients (P=0.014 and P<0.001 respectively). Assessment of LVI and BVI by IHC, using D2-40 and Factor VIII, improves prediction of outcome in patients with node negative and triple negative breast cancer and was superior to the conventional detection method. Breast cancer is recognised as a complex molecular disease and histologically identical tumours may have highly variable outcomes, including different responses to therapy. Therefore, there is a compelling need for new prognostic and predictive markers helpful of selecting patients at risk and patients with aggressive diseases who might benefit from adjuvant and targeted therapy. It is increasingly recognised that the development and progression of human breast cancer is not only determined by genetically abnormal cells, but also dependent on complex interactions between malignant cells and the surrounding microenvironment. This has led to reconsider the features of tumour microenvironment as potential predictive and prognostic markers. Among these markers, tumour stroma percentage (TSP) and tumour budding, as well as local tumour inflammatory infiltrate have received recent attention. In particular, the local environment of cytokines, proteases, angiogenic and growth factors secreted by inflammatory cells and stromal fibroblasts has identified crucial roles in facilitating tumour growth, and metastasis of cancer cells through lymphatic and/or blood vessel invasion. This might help understand the underlying process promoting tumour invasion into these vessels. An increase in the proportion of tumour stroma and an increase in the dissociation of tumour cells have been associated with poorer survival in a number of solid tumours, including breast cancer. However, the interrelationship between these variables and other features of the tumour microenvironment in different subgroups of breast cancer are not clear. Also, whether their prognostic values are independent of other components of the tumour microenvironment have yet to be identified. Therefore, the relationship between TSP, clinicopathological characteristics and outcome in patients with invasive ductal breast cancer, in particular node negative and triple negative disease was examined in patients with invasive ductal breast cancer (n=361). The TSP was assessed on the haematoxylin and eosin-stained tissue sections. With a cut-off value of 50% TSP, patients with ≤50% stroma were classified as the low-TSP group and those with >50% stroma were classified as the high-TSP group. A total of 109 (30%) patients had high TSP. Patients with high TSP were old age (P=0.035), had involved lymph node (P=0.049), Her-2 positive tumours (P=0.029), low-grade peri-tumoural inflammatory infiltrate (P=0.034), low CD68+ macrophage infiltrate (P<0.001), low CD4+ (P=0.023) and low CD8+ T-lymphocytes infiltrate (P=0.017), tumour recurrence (P=0.015) and shorter CSS (P<0.001). In node negative patients (n=207), high TSP was associated with low CD68+ macrophage infiltrate (P=0.001), low CD4+ (P=0.040) and low CD8+ T-lymphocytes infiltrate (P=0.016) and shorter CSS (P=0.005). In triple negative patients (n=103), high TSP was associated with increased tumour size (P=0.017) high tumour grade (P=0.014), low CD8+ T-lymphocytes infiltrate (P=0.048) and shorter CSS (P=0.041). The 15-year cancer specific survival rate was 79% vs 21% in the low-TSP group vs high-TSP group. On multivariate survival analysis, a high TSP was associated with reduced CSS in the whole cohort (P=0.007), node negative patients (P=0.005) and those who received systemic adjuvant therapy (P=0.016), independent of other pathological characteristics including local host inflammatory responses. Therefore, a high TSP in invasive ductal breast cancer was associated with recurrence and poorer long-term survival. The inverse relation with the tumour inflammatory infiltrate highlights the importance of the amount of tumour stroma on immunological response in patients with invasive ductal breast cancer. Implementing this simple and reproducible parameter in routine pathological examination may help optimise risk stratification in patients with breast cancer. Similarly, the relationship between tumour budding, clinicopathological characteristics and outcome was examined in patients with invasive ductal breast cancer (n=474), using routine pathological sections. Tumour budding was associated with several adverse pathological characteristics, including positive lymph node (P=0.009), presence of LVI (P<0.001), and high TSP (P=0.001) and low-grade general peri-tumural inflammatory infiltrative (P=0.002). In node negative patients, a high tumour budding was associated with presence of LVI (P<0.001) and low-grade general peri-tumural inflammatory infiltrative (P=0.038). On multivariate survival analysis, tumour budding was associated with reduced CSS (P=0.001), independent of nodal status, tumour necrosis, CD8+ and CD138+ inflammatory cells infiltrate, LVI, BVI and TSP. Furthermore, tumour budding was independently associated with reduced CSS in node negative patients (P=0.004) and in those who have low TSP (P=0.003) and high-grade peri-tumoural inflammatory infiltrative (P=0.012). A high tumour budding was significantly associated with shorter CSS in luminal B and triple negative breast cancer subtypes (all P<0.001). Therefore, tumour budding was a significant predictor of poor survival in patients with invasive ductal breast cancer, independent of adverse pathological characteristics and components of tumour microenvironment. These results suggest that tumour budding may promote disease progression through a direct effect on local and distant invasion into lymph nodes and lymphatic vessels. Therefore, detection of tumour buds at the stroma invasive front might therefore represent a morphologic link between tumour progression, lymphatic invasion, spread of tumour cells to regional lymph nodes, and the establishment of metastatic dissemination. Given the potential importance of the tumour microenvironment, the characterisation of intracellular signalling pathways is important in the tumour microenvironment and is of considerable interest. One plausible signalling molecule that links tumour stroma, inflammatory cell infiltrate and tumour budding is the signal transducer and activator of transcription (STAT). The relationship between total and phosphorylated STAT1 (ph-STAT1), and total and ph-STAT3 tumour cell expression, components of tumour microenvironment and survival in patients with invasive ductal breast cancer was examined. IHC of total and ph-STAT1/STAT3 was performed on tissue microarray of 384 breast cancer specimens. Cellular STAT1 and cellular STAT3 expression at both cytoplasmic and nuclear locations were combined and identified as STAT1/STAT3 tumour cell expression. These results were then related to CSS and phenotypic features of the tumour and host. A high ph-STAT1 and a high ph-STAT3 tumour cell expression was associated with increased ER (P=0.001 and P<0.001 respectively) and PR (all P<0.05), reduced tumour grade (P=0.015 and P<0.001 respectively) and necrosis (all P=0.001). Ph-STAT1 was associated with increased general peri-tumoural inflammatory infiltrate (P=0.007) and ph-STAT3 was associated with lower CD4+ T-lymphocyte infiltrate (P=0.024). On multivariate survival analysis, including both ph-STAT1 and ph-STAT3 tumour cell expression, only high ph-STAT3 tumour cell expression was significantly associated with improved CSS (P=0.010) independent of other tumour and host-based factors. In patients with high necrosis grade, high ph-STAT3 tumour cell expression was independent predictor of improved CSS (P=0.021). Ph-STAT1 and ph-STAT3 were also significantly associated with improved cancer specific survival in luminal A and B subtypes. STAT1 and STAT3 tumour cell expression appeared to be an important determinant of favourable outcome in patients with invasive ductal breast cancer. The present results suggest that STATs may affect disease outcome through direct impact on tumour cells, and the surrounding microenvironment. The above observations of the present thesis point to the importance of the tumour microenvironment in promoting tumour budding, LVI and BVI. The observations from STATs work may suggest that an important driving mechanism for the above associations is the presence of tumour necrosis, probably secondary to hypoxia. Further work is needed to examine the interaction of other molecular pathways involved in the tumour microenvironment, such as HIF and NFkB in patients with invasive ductal breast cancer.