Equivariant neural networks for inverse problems

In recent years the use of convolutional layers to encode an inductive bias (translational equivariance) in neural networks has proven to be a very fruitful idea. The successes of this approach have motivated a line of research into incorporating other symmetries into deep learning methods, in the form of group equivariant convolutional neural networks. Much of this work has been focused on roto-translational symmetry of R ^d , but other examples are the scaling symmetry of R ^d and rotational symmetry of the sphere. In this work, we demonstrate that group equivariant convolutional operations can naturally be incorporated into learned reconstruction methods for inverse problems that are motivated by the variational regularisation approach. Indeed, if the regularisation functional is invariant under a group symmetry, the corresponding proximal operator will satisfy an equivariance property with respect to the same group symmetry. As a result of this observation, we design learned iterative methods in which the proximal operators are modelled as group equivariant convolutional neural networks. We use roto-translationally equivariant operations in the proposed methodology and apply it to the problems of low-dose computerised tomography reconstruction and subsampled magnetic resonance imaging reconstruction. The proposed methodology is demonstrated to improve the reconstruction quality of a learned reconstruction method with a little extra computational cost at training time but without any extra cost at test time.

Learning the Sampling Pattern for MRI

The discovery of the theory of compressed sensing brought the realisation that many inverse problems can be solved even when measurements are "incomplete". This is particularly interesting in magnetic resonance imaging (MRI), where long acquisition times can limit its use. In this work, we consider the problem of learning a sparse sampling pattern that can be used to optimally balance acquisition time versus quality of the reconstructed image. We use a supervised learning approach, making the assumption that our training data is representative enough of new data acquisitions. We demonstrate that this is indeed the case, even if the training data consists of just 7 training pairs of measurements and ground-truth images; with a training set of brain images of size 192 by 192, for instance, one of the learned patterns samples only 35% of k-space, however results in reconstructions with mean SSIM 0.914 on a test set of similar images. The proposed framework is general enough to learn arbitrary sampling patterns, including common patterns such as Cartesian, spiral and radial sampling.

CA 15–3: A Prognostic Marker in Breast Cancer

CA 15–3 (also known as MUC1) is the most widely used serum marker in breast cancer. MUC1 is a large transmembrane glycoprotein which is frequently overexpressed and aberrantly glycosylated in cancer. Physiologically, MUC1 appears to play a role in cell adhesion and the high levels present in cancer may be causally involved in metastasis. At present the main uses of CA 15–3 are in preclinically detecting recurrent breast cancer and monitoring the treatment of patients with advanced breast cancer. In a prospective study of 368 patients we show that patients with high preoperative levels of CA 15–3 (>30.4 U/mL) had a worse outcome than patients with low levels of the marker. In multivariate analysis CA 15–3 as a prognostic marker was independent of both tumor size and nodal status. Furthermore, in multivariate analysis the prognostic impact of CA 15–3 was stronger than that of tumor size and at least as strong as nodal status. CA 15–3 may thus be the first independent prognostic serum marker in breast cancer.

The clinical value of CEA and CA 15-3 in breast cancer management

The value of tumour-associated antigens CEA and CA 15-3 was studied in patients with breast cancer over a 4-year period. A total of 252 patients with primary or recurrent disease had available and corresponding CEA and CA 15-3 values at diagnosis and during follow-up and were studied in detail. Preoperative and three-monthly serial postoperative levels were measured in each patient. Ten of 11 patients presenting with primary and concurrent metastatic disease had elevated CA 15-3 levels (> 25 I.U./ml) as compared to 6 with CEA (> 5 ng/ml). Fourty-seven patients developed locoregional recurrence of which 15 had concurrent metastatic disease. CA 15-3 was elevated in 14 cases while CEA in 11. Of 32 patients with locoregional recurrence alone, 18 later developed metastatic disease at a mean follow-up time of 17.5 months. There was a significant correlation between CA 15-3 value at locoregional recurrence and time to subsequent metastasis (r = 0. -0.57, P = 0.0133). CEA was elevated in 64%, CA 15-3 in 87% and either marker in 94% of 87 patients diagnosed with metastatic disease. Of 53 patients with serial markers and metastatic disease, 72% (38/53) had rising CA 15-3 levels prior to diagnosis with a mean lead time of 9.9 months. Use of CEA in conjunction improved lead time detection to 83%.

This study demonstrates that CA 15-3 is superior to CEA at detecting metastatic disease at initial presentation and during follow-up. Use of CEA in conjunction with CA 15-3 improves the detection of systemic disease.

Preoperative CA 15-3 concentrations predict outcome of patients with breast carcinoma

BACKGROUND

CA 15-3 is a breast-associated mucin that is elevated in the majority of breast carcinoma patients with distant metastases. Currently, the main application of this marker is in monitoring and detecting recurrences in patients with diagnosed breast carcinoma.

METHODS

Preoperative serum concentrations (prior to excision of the primary tumor) of CA 15-3 were measured in 368 patients undergoing potentially curative surgical treatment for early breast carcinoma. These results were compared with prospectively recorded clinicopathologic characteristics and patient outcome data.

RESULTS

A weak but significant positive association was found between CA 15-3 concentrations and both tumor stage and the number of involved axillary lymph nodes but not between CA 15-3 concentrations and estrogen receptor status. Patients with high concentrations of CA 15-3 had a significantly worse prognosis than patients with low concentrations. Using an optimum cutoff value of 30.38 U/mL, the probability of disease free survival at 5 years was 44% in patients with high CA 15-3 levels compared with 65% in patients with low CA 15-3 levels (P = 0.002, Mantel-Cox log rank test). The corresponding probabilities for overall survival were 67% and 83%, respectively (P < 0.001). The association of preoperative CA 15-3 levels with outcome was maintained in multivariate survival analysis and was not explained by the association between CA 15-3 and tumor size or lymph node burden. The relation between CA 15-3 and outcome also was found within some patient subgroups identified by traditional prognostic factors (axillary lymph node positive patients, patients with primary tumors >2 cm in greatest dimension, and patients with estrogen receptor positive tumors).

CONCLUSIONS

Preoperative serum concentrations of CA 15-3 appear to have a significant relation to outcome in patients with early breast carcinoma and may have a role in the rational selection of patients for appropriate adjuvant treatments. To the authors' knowledge, CA 15-3 thus is one of the first circulating markers shown to be an independent prognostic indicator in patients with breast carcinoma.

CA 15-3 in patients with locoregional and metastatic breast carcinoma

BACKGROUND

The value of circulating CA 15-3 levels was assessed in 129 patients with recurrent breast carcinoma.

METHODS

Patients were divided into four subgroups, according to the following: Group A, locoregional recurrence alone; Group B, locoregional and subsequent systemic recurrence; Group C, combined locoregional and systemic recurrence; and Group D, differing sites of systemic disease.

RESULTS

One of 14 patients with locoregional disease alone had increased levels of CA 15-3 (> 25 U/ml). However, 96% of patients (22 of 23 patients) with combined local and systemic disease had increased tumor marker levels. The difference in CA 15-3 levels in patients with combined disease compared with patients with local disease alone was statistically significant (117.0 versus 17.5 U/ml, respectively; P < 0.02). Twenty-four patients with locoregional recurrence later had distant metastasis develop. In this group, patients with an increased CA 15-3 value had a significantly shorter lead time to the development of distant metastases compared with patients with normal tumor marker levels (20.8 +/- 3.3 versus 10.3 +/- 2.7 months, respectively; P < 0.03). CA 15-3 values at diagnosis were increased in 88% of 115 patients with metastatic disease. There was no significant difference in CA 15-3 levels among metastases to lung, liver, and bone nor was there any difference between single and multiple sites of distant metastasis. CA 15-3 is an excellent marker for systemic recurrence of breast carcinoma.

CONCLUSIONS

Increased levels and no clinical evidence of recurrence strongly indicate the presence of occult metastatic disease.