Detecting changes in retinal function: Analysis with Non-Stationary Weibull Error Regression and Spatial enhancement (ANSWERS)

Abstract

Visual fields measured with standard automated perimetry are a benchmark test for determining retinal function in ocular pathologies such as glaucoma. Their monitoring over time is crucial in detecting change in disease course and, therefore, in prompting clinical intervention and defining endpoints in clinical trials of new therapies. However, conventional change detection methods do not take into account non-stationary measurement variability or spatial correlation present in these measures. An inferential statistical model, denoted 'Analysis with Non-Stationary Weibull Error Regression and Spatial enhancement' (ANSWERS), was proposed. In contrast to commonly used ordinary linear regression models, which assume normally distributed errors, ANSWERS incorporates non-stationary variability modelled as a mixture of Weibull distributions. Spatial correlation of measurements was also included into the model using a Bayesian framework. It was evaluated using a large dataset of visual field measurements acquired from electronic health records, and was compared with other widely used methods for detecting deterioration in retinal function. ANSWERS was able to detect deterioration significantly earlier than conventional methods, at matched false positive rates. Statistical sensitivity in detecting deterioration was also significantly better, especially in short time series. Furthermore, the spatial correlation utilised in ANSWERS was shown to improve the ability to detect deterioration, compared to equivalent models without spatial correlation, especially in short follow-up series. ANSWERS is a new efficient method for detecting changes in retinal function. It allows for better detection of change, more efficient endpoints and can potentially shorten the time in clinical trials for new therapies.

Figure 1. Visual field measured by standard automated perimetry (SAP).

(a) Contrast stimulus from SAP is projected on different locations of retina. The response from subject is captured when the stimulus is perceived. (b) SAP assesses differential light sensitivity (DLS) of the retina and corresponding visual pathway. (c) DLSs are measured at various locations (dots) on the retina. The point (0°,0°) indicates central vision that corresponds to the fovea on the retina. Optic nerve head is the anatomical blind spot. The test locations are not only correlated to their neighbours but also by the optic nerve fibres (some of which are shown as blue curves) passing through them. The whole visual field can be divided into superior and inferior hemifields on vertical and nasal and temporal regions on horizontal. (d) The DLS at a location on the retina is derived at the 50% probability of the visual system responding to a contrast stimulus and is related to the biological response to light of relay neurones in the visual pathway. (e) The DLS is measured in log scale, which in Humphrey Field Analyzer (Carl Zeiss Meditec Inc, Dublin, CA, USA) is calculated as dB =  where is the luminance of the stimulus in apostilbs and 31.6 apostilbs is the background luminance. The DLS ranges between 0 dB (high contrast stimulus, blindness) and around 35 dB (low contrast stimulus, healthy) and is displayed as a conventional gray-scale plot. Darker shading represents lower DLS. (f) Measurements of DLS over time form a complex spatial-temporal time series.

Figure 2. Spatial correlation between each location and all other locations in the visual field.

The composition of the graph is a 24-2 visual field as shown in Figure 1c. At each visual field location, an image, with the shape of a 24-2 visual field, represents the correlation between this location and all locations in the visual field. The grayscale bar, shown at the location of the blind spot, indicates the level of correlation.

Figure 3. Histograms of retest differential light sensitivities at levels between 0

The derived probability density function of the Weibull mixture is superimposed in red.

Figure 4. Examples comparing ANSWER and ordinary linear regression.

The retest distributions of corresponding differential light sensitivity measurements are superimposed as grey areas. The scored probability densities by the ANSWER regression line are marked on the retest distributions.

Figure 5. Time to detect deterioration for linear regression of mean deviation (MD), point-wise linear regression (PLR), ANSWERS and ANSWER at false positive rates between 0 and 15%.

The number of contiguous points in point-wise linear regression are shown in the square points.

Figure 6. The hit rates of linear regression of mean deviation (MD), point-wise linear regression (PLR), ANSWERS and ANSWER with series lengths (length) of 5, 7, 9 and 11.

The number of contiguous points in point-wise linear regression are shown in the square points. The hit rates are estimated at false positive rates between 0 and 15%.