Glaucoma is an optic neuropathy characterized by the progressive loss of retinal ganglion cells (RGCs),
1 which exhibit structural and functional changes prior to their death by apoptosis.
2 Early RGC structural changes include reduction in the length and number of dendrites and the area of the dendritic arbor, as seen from mouse models of optic nerve injury.
3,4 Functional changes in the RGCs prior to apoptosis can include an increased firing rate from increased excitability
5,6 and a fall in the mean and peak spike rates.
7 Detecting these early functional changes might potentially aid the prediction of future RGC loss in glaucoma and thus enhance the clinical management of progression of this condition, which was estimated to affect 64.3 million people in the world between the ages of 30 and 80 years in 2013.
8
Electrophysiological recordings from the eye may be able to detect the earliest changes in RGC function. In a study on glaucoma suspects, Banitt et al.
9 showed that a 10% change in the pattern electroretinogram (PERG) amplitude preceded the same change in peripapillary retinal nerve fiber layer (RNFL) thickness by 8 years. Liu et al.
10 worked in a rat model and showed that, with chronic intraocular pressure (IOP) elevation, the positive scotopic threshold response was reduced by 25% in animals with elevated pressure without any changes in the optical coherence tomography (OCT) parameters and that these changes reversed with normalization of the IOP. It is therefore plausible that the electroretinogram (ERG) is a tool that could be used to capture such early functional abnormalities of the RGCs. The ERG is a time-domain signal of the electrical activity of the retina in response to light stimuli. When recorded under light-adapted conditions with a brief presentation of a red-on-blue stimulus, a slow negative going wave occurring after an initial a-wave trough and b-wave peak—termed the photopic negative response (PhNR)—is typically observable.
11 The PhNR is a response that arises from the RGCs and has been demonstrated to be reduced in glaucoma, both in experimental models and in clinical studies.
12–17 It can show improvement in patients with glaucoma when the IOP is lowered
18 and has been used as a marker of inner retinal function in patients with glaucoma when treated with nicotinamide.
19 The signal between the b-wave peak and the PhNR trough is frequently interrupted by the i-wave, which is thought to arise from the cells distal to the RGCs.
20 If the i-wave is present, the PhNR can be recorded as either the negative wave between the b-wave and i-wave (the PhNR1) or the negative wave following the i-wave (the PhNR2). We have previously shown that a combination of key amplitude markers of the ERG—namely, the a-wave, b-wave, i-wave, and PhNR (PhNR1 and PhNR2) amplitudes—better predicts glaucoma severity than the PhNR amplitude alone.
21 These findings underscore how a photopic ERG contains more information about RGC function than is captured by conventional PhNR measures alone.
The ERG measured from a single differential pair electrode is a mixture of the underlying processes, which occur at different times after the stimulus.
22 The underlying mechanisms are complex
23 with feedback
24 and feedforward pathways,
25,26 and there are slow and fast frequency components from the different sources of the signal. Time–frequency analysis has the potential to extract these components from the photopic ERG to provide further insights into RGC function, above and beyond what is captured by the amplitude markers of the ERG. With time–frequency analysis, a vector of amplitude measurements is transformed into a matrix of coefficients with axes of time and frequency so that the magnitude and timing of frequency components within the signal can be determined. The wavelet transform, a form of time–frequency analysis, has been described as an alternative to short-time Fourier transform.
27,28 A wavelet is a small function (little wave) that acts as a filter and can localize energy within the signal in time and frequency. This is achieved through multiplication of the signal and wavelet after translation and dilation of the wavelet relative to the signal for time and frequency resolution, respectively. Wavelets have been applied to the PERG,
29 multifocal ERG,
30 and photopic ERG
31 in the assessment of glaucoma.
There are two types of wavelet transforms: continuous wavelet transform (CWT)
28 and discrete wavelet transform (DWT).
32
The CWT is obtained by convoluting the wavelet with the signal for all values of scale (e.g., frequency) and time lag, creating a continuous scalogram showing the energy of the signal at each frequency and time point. However, this creates a highly redundant output matrix, as both its width and length are now equal to the number of samples in the original time series vector. Forte and colleagues
33 showed that the Morlet wavelet could be used to isolate oscillatory potentials in rat ERGs. Behbahani and colleagues
34 used a Mexican hat wavelet to determine the dominant frequencies associated with the PhNR in patients with central retinal vein occlusion and found that the dominant frequency decreased. However, the CWT has not thus far found clinical utility for glaucoma.
On the other hand, the DWT
32,35,36 uses set scales and time lags at discrete values, where the output is in the form of a binary tree and the total number of coefficients is equal to the number of samples in the input, which necessarily must be of length equal to a power of two. Specifically, it uses a low-pass filter (scaling function) and a high-pass filter (wavelet function), followed by downsampling by two. There are many mother wavelets available for the DWT. Selection of the mother wavelet optimizes the resolution in time and frequency for the temporal and spectral content of the signal. Various techniques for mother wavelet selection have been described,
37 including minimizing informational cost. The DWT has been applied to glaucoma, but previous studies have generally used a single feature derived from the transform.
30,38 A recent paper explored the use of the DWT in the photopic ERG and found differences between individuals diagnosed with autism spectrum disorder compared with control subjects.
39
A different approach to time–frequency analysis is the matching pursuit (MP) algorithm.
40 The MP algorithm decomposes a time-domain signal into a linear combination of subsignals of the same length termed atoms. The full set of atoms is referred to as the dictionary. The decomposition output is a set consisting of a coefficient for each atom in the dictionary. The process of generating the dictionary usually begins with a discrete wavelet family: small vectors representing digital filters.
32 In contrast to the DWT, the atoms in the MP decomposition are padded to the length of the signal rather than a much shorter vector that slides along the signal. The dictionary thus includes all of the time shifts of a given wavelet as separate atoms. The algorithm process is known as “greedy.” It starts by finding the best match to the signal from the entire dictionary, removes that part of the signal, and then finds the next best match and so on. The number of iterations is usually equal to the length of the dictionary, although a smaller fixed number can be used, or the algorithm can reach a stopping criterion. The result of the transformation is the coefficient for each atom and the index into the dictionary used. The dimensionality of the transformation may be larger or smaller than the original signal.
We therefore posed this optimization problem: Given a continuous outcome measure (the estimated RGC), can additional features extracted by the DWT or the MP informing linear or MARS models yield better performance than time-domain amplitude features alone? Although the DWT is more straightforward for others to replicate with widely available software, the MP technique may offer better time localization for low frequencies. Some studies have compared the two techniques in hyperspectral imaging
41 and electroencephalogram analysis,
42 although these studies addressed classification rather than regression problems and the differences were modest. Both techniques in theory allow the extraction of multiple features from the underlying processes and could plausibly better characterize the extent of retinal dysfunction than time-domain features alone.
Given the ability of the DWT and MP algorithms to extract novel time–frequency features from the photopic ERG, we examined whether these approaches, when used together with an extended set of amplitude markers, could be used to better predict glaucoma severity (or the extent of RGC loss and dysfunction). We compared the incremental benefit of each method and the combination of both. Our aim was not to develop a new clinical tool for the diagnosis of glaucoma or the classification of its severity, as clinicians currently have OCT, standard automated perimetry (SAP), and clinical examination for that. Rather, the aim of this study was to elucidate additional information within the ERG that can be extracted by time–frequency techniques and which might ultimately be useful in, for example, building a predictive model of progression.