Fractional averaging theory for discrete fractional-order system with impulses.

In this paper, we improve the averaging theory on both finite and infinite time intervals for discrete fractional-order systems with impulses. By employing new techniques, generalized impulsive discrete fractional-order Gronwall inequality is introduced. In addition, the closeness of solutions for the discrete fractional-order systems with impulses and the averaged discrete fractional-order systems with impulses is derived. Finally, three examples are provided to illustrate the efficiency of our main results.

Separation and thickness measurements of superficial and deep slabs of the retinal nerve fiber layer in healthy and glaucomatous eyes.

Purpose: Describe a new method to analyze retinal nerve fiber layer (RNFL) thickness maps. Design: Cross-sectional study. Subjects: RNFL thickness maps of healthy and glaucomatous eyes. Methods: Optical coherence tomography (OCT) RNFL raster scans from 98 healthy and 50 glaucomatous eyes were analyzed. The RNFL thickness maps were separated into superficial (SNFL) and deep (DNFL) slabs through a partial thickness plane set at the modal thickness (mode). Association between mode and OCT signal strength (SS), age, axial length, and visual field mean deviation (VFMD) was tested (Pearson coefficient, r). Thicknesses of inferior and superior SNFL regions (i-,s-SNFL), and inferior, superior, nasal, and temporal DNFL regions (i-,s-,n-,t-DNFL) were calculated. The regions thicknesses were compared between healthy and glaucomatous eyes (t-test) and between glaucomatous eyes with early, moderate, and severe disease (ANOVA and linear regressions of thickness on VFMD). Diagnostic accuracy and correlation with VFMD of RNFL regions thicknesses were calculated as the area under the receiver operating characteristic curve (AUC) and Pearson r, respectively. P<0.05 was considered significant. Main outcome: Thickness of regions in SNFL and DNFL slabs. Results: The mode was not associated with SS, age, axial length, or VFMD, it circumscribed the thicker RNFL around the optic disc of healthy and glaucomatous eyes, and it was used to separate the SNFL and DNFL slabs of RNFL thickness maps. The thickness of the SNFL slab was less in glaucomatous eyes than in healthy eyes (P<0.001). S-SNFL and i-SNFL thicknesses (respectively) were 86.0±8.2µm and 87.3±9.6µm in healthy eyes vs. 66.1±9.1µm and 63.4±8.2µm in glaucomatous eyes (P<0.001 for both). The thickness of the DNFL slab was similar between groups (P=0.19). T-DNFL thickness was 37.0±5.3µm in healthy eyes vs. 33.9±5.0µm in glaucomatous eyes (P<0.001); thicknesses of all other DNFL regions were similar. The SNFL regions only thinned with progressively worse glaucoma severity, had excellent AUCs (AUC≥0.95, P<0.001), and correlated strongly with VFMD (r≥0.60, P<0.001). Conclusions: Glaucomatous RNFL thinning is predominantly detected within a slab with thickness greater than the mode. SNFL thickness has great AUC and correlation with VFMD in glaucomatous eyes. The usefulness for diagnosis and monitoring of glaucoma needs further study.

Comparison of Matrix with Humphrey Field Analyzer II with SITA.

PURPOSE: To study the performance of the Matrix perimeter compared with the Humphrey Field Analyzer II (HFA) with the Swedish Interactive Thresholding Algorithm over the range of contrast sensitivities each machine could estimate. METHODS: Fifty stable glaucoma subjects at various stages of disease and three normal subjects had visual fields testing done on five different days within 8 weeks with both perimeters. Intraclass correlation coefficient of mean deviation, pattern standard deviation, and the SD of repeat measurements were evaluated. The repeatability of the sensitivity estimates at individual locations and global indices was quantified, as well as their dependence on disease severity. The relationship between sensitivity determinations with the two instruments was explored (principal curve analysis). RESULTS: Mean deviation on the HFA ranged from -31 to +2.5 dB. The mean deviation and pattern standard deviation had intraclass correlation coefficients above 0.90 for both instruments. Over most of the useful range (above 20 dB on the HFA), a difference of 1 dB for the Matrix corresponded to a difference of 2 dB for the HFA. The SD of repeat measurements increased with disease severity with HFA, but not with Matrix, except that values of 12 or 34 dB were highly variable on repeat. Variability was reduced for both HFA and Matrix when duplicate sensitivity values were used. A single Matrix test provided only 15 possible sensitivity values, unevenly spaced, but the average of duplicate measurements provided more numerous sensitivity values. A learning effect was detected for Matrix. CONCLUSIONS: The decibel values reported by the two machines are not equivalent. Variability of sensitivity determinations is affected more by the sensitivity level with HFA than with Matrix. Duplicate measurements for baseline and follow-up evaluation could be important, especially for Matrix. Further information on learning effects is needed, as is commercially available progression software for Matrix.

Longitudinal changes in peripapillary atrophy in the ocular hypertension treatment study: a case-control assessment.

PURPOSE: To explore the association between peripapillary atrophy (PPA) area and conversion from ocular hypertension (OHT) to glaucoma. DESIGN: Prospective, longitudinal cohort study of cases and controls. PARTICIPANTS: We included 279 age-matched and follow-up time-matched eyes with OHT that converted to glaucoma and 279 eyes with OHT that did not convert to glaucoma. METHODS: Initial and last acceptable optic disc photos were analyzed. Disc, α-zone, and ß-zone PPA were traced independently by 2 trained readers and their areas were measured with Photoshop. The α-zone and ß-zone areas were expressed as a percentage of optic disc area. MAIN OUTCOME MEASURES: α-Zone and ß-zone PPA size over time. RESULTS: Intraclass correlation coefficients (ICCs) demonstrated that readers had good agreement on disc area (ICC = 0.97) and ß-zone (ICC = 0.82), but not α-zone (ICC = 0.48). The ß-zone, as a percentage of disc area, increased in size (P < 0.001) in both eyes with incident primary open-angle glaucoma (mean, 10.6%; standard deviation, 22.6%) and matched controls (mean, 10.1%; standard deviation, 33.7) over follow-up (mean, 12.3 years). The increase in size did not differ between cases and controls (P = 0.82). Enlargement of the ß-zone was not correlated with follow-up time (P = 0.39). CONCLUSIONS: The results did not show a difference in size of the ß-zone at baseline between eyes that proceed to develop glaucoma and those that do not. Moreover, the ß-zone enlarges equally in case and control eyes during follow-up.

Estimating the rate of retinal ganglion cell loss in glaucoma.

PURPOSE: To present and evaluate a new method of estimating rates of retinal ganglion cell (RGC) loss in glaucoma by combining structural and functional measurements. DESIGN: Observational cohort study. METHODS: The study included 213 eyes of 213 glaucoma patients followed up for an average of 4.5 ± 0.8 years with standard automated perimetry visual fields and optical coherence tomography. A control group of 33 eyes of 33 glaucoma patients underwent repeated tests over a short period to test the specificity of the method. An additional group of 52 eyes from 52 healthy subjects followed up for an average of 4.0 ± 0.7 years was used to estimate age-related losses of RGCs. Estimates of RGC counts were obtained from standard automated perimetry and optical coherence tomography, and a weighted average was used to obtain a final estimate of the number of RGCs for each eye. The rate of RGC loss was calculated for each eye using linear regression. Progression was defined by a statistically significant slope faster than the age-expected loss of RGCs. RESULTS: From the 213 eyes, 47 (22.1%) showed rates of RGC loss that were faster than the age-expected decline. A larger proportion of glaucomatous eyes showed progression based on rates of RGC loss rather than based on isolated parameters from standard automated perimetry (8.5%) or optical coherence tomography (14.6%; P < .01), while maintaining similar specificities in the stable group. CONCLUSIONS: The rate of RGC loss estimated from combining structure and function performed better than either isolated structural or functional measures for detecting progressive glaucomatous damage.

Progression of patterns (POP): a machine classifier algorithm to identify glaucoma progression in visual fields.

PURPOSE: We evaluated Progression of Patterns (POP) for its ability to identify progression of glaucomatous visual field (VF) defects. METHODS: POP uses variational Bayesian independent component mixture model (VIM), a machine learning classifier (MLC) developed previously. VIM separated Swedish Interactive Thresholding Algorithm (SITA) VFs from a set of 2,085 normal and glaucomatous eyes into nine axes (VF patterns): seven glaucomatous. Stable glaucoma was simulated in a second set of 55 patient eyes with five VFs each, collected within four weeks. A third set of 628 eyes with 4,186 VFs (mean ± SD of 6.7 ± 1.7 VFs over 4.0 ± 1.4 years) was tested for progression. Tested eyes were placed into suspect and glaucoma categories at baseline, based on VFs and disk stereoscopic photographs; a subset of eyes had stereophotographic evidence of progressive glaucomatous optic neuropathy (PGON). Each sequence of fields was projected along seven VIM glaucoma axes. Linear regression (LR) slopes generated from projections onto each axis yielded a degree of confidence (DOC) that there was progression. At 95% specificity, progression cutoffs were established for POP, visual field index (VFI), and mean deviation (MD). Guided progression analysis (GPA) was also compared. RESULTS: POP identified a statistically similar number of eyes (P > 0.05) as progressing compared with VFI, MD, and GPA in suspects (3.8%, 2.7%, 5.6%, and 2.9%, respectively), and more eyes than GPA (P = 0.01) in glaucoma (16.0%, 15.3%, 12.0%, and 7.3%, respectively), and more eyes than GPA (P = 0.05) in PGON eyes (26.3%, 23.7%, 27.6%, and 14.5%, respectively). CONCLUSIONS: POP, with its display of DOC of progression and its identification of progressing VF defect pattern, adds to the information available to the clinician for detecting VF progression.

Glaucoma Progression Analysis software compared with expert consensus opinion in the detection of visual field progression in glaucoma.

PURPOSE: To compare the results of Glaucoma Progression Analysis (GPA, Carl Zeiss Meditec, Dublin, CA) to subjective expert consensus in the detection of glaucomatous visual field progression. DESIGN: Retrospective, observational case series. PARTICIPANTS: We included 100 eyes of 83 glaucoma patients. METHODS: Five serial Humphrey visual fields from 100 eyes of 83 glaucoma patients were evaluated by 5 masked glaucoma subspecialists for determination of progression. Four months later, with a randomly reordered patient sequence, the same visual field series were reevaluated by the same graders, at which time they had access to the Glaucoma Progression Analysis (GPA) printout. MAIN OUTCOME MEASURES: The level of agreement between majority expert consensus and GPA, both before and after access to GPA data, was assessed using kappa statistics. RESULTS: On initial review and on reevaluation with access to the GPA printout, the level of agreement between majority expert consensus and GPA was fair (kappa = 0.52, 95% confidence interval [CI], 0.35-0.69 and kappa = 0.62; 95% CI, 0.46-0.78, respectively). Expert consensus was more likely to classify a series of fields as showing progression than was GPA (P ≤ 0.002). There was good agreement between expert consensus on initial review and reevaluation 4 months later (kappa = 0.77; 95% CI, 0.65-0.90). CONCLUSIONS: The level of agreement between majority expert consensus of subjective determination of visual field progression and GPA is fair. In cases of disagreement with GPA, the expert consensus classification was usually progression. Access to the results of GPA did not significantly change the level of agreement between expert consensus and the GPA result; however, expert consensus did change in 11 of 100 cases.

Use of colour Doppler imaging in ocular blood flow research.

The main objective of this report is to encourage consistent quality of testing and reporting within and between centres that use colour Doppler imaging (CDI) for assessment of retrobulbar blood flow. The intention of this review is to standardize methods in CDI assessment that are used widely, but not to exclude other approaches or additional tests that individual laboratories may choose or continue to use.

The first signs of glaucomatous cupping in the optic nerve.

Evaluation of the optic disc is important for both the diagnosis of glaucoma, and in monitoring the progress of glaucoma. Along with visual field examination, it allows the presence of glaucoma to be recognized, and for progressive damage to be seen. Glaucoma can occur despite intraocular pressure (IOP) in the normal range, but as importantly, can be absent even when the IOP is distinctly high.

Quantifying discordance between structure and function measurements in the clinical assessment of glaucoma.

OBJECTIVE: To evaluate a new method of quantifying and visualizing discordance between structural and functional measurements in glaucomatous eyes by predicting the visual field (VF) from retinal nerve fiber layer thickness (RNFLT) using a bayesian radial basis function. METHODS: Five GDx VCC RNFLT scans and 5 Humphrey 24-2 Swedish Interactive Thresholding Algorithm VF tests were performed for 50 glaucomatous eyes from 50 patients. A best-available estimate (BAE) of the true VF was calculated as the pointwise median of these 5 replications. This BAE VF was compared with every RNFLT-predicted VF from the bayesian radial basis function and every measured VF. Predictability of VFs from RNFLT was established from previous data. A structure-function pattern discordance map and a structure-function discordance index (scores of 0-1) were established from the predictability limits for each structure-function measurement pair to quantify and visualize the discordance between the structure-predicted and measured VFs. RESULTS: The mean absolute difference between the structure-predicted and BAE VFs was 3.9 dB. The mean absolute difference between measured and BAE VFs was 2.6 dB. The mean (SD) structure-function discordance index score was 0.34 (0.11). Ninety-seven (39%) of the structure-predicted VFs showed significant discordance (structure-function discordance index score >0.3) from measured VFs. CONCLUSIONS: On average, the bayesian radial basis function predicts the BAE VF from RNFLT slightly less well than a measured VF from the 5 VFs composing the BAE VF. The pattern discordance map highlights locations with structure-function discordance, with the structure-function discordance index providing a summary index. These tools may help clinicians trust the mutually confirmatory structure-function measurements with good concordance or identify unreliable ones with poor concordance.

Comparison of automated analysis of Cirrus HD OCT spectral-domain optical coherence tomography with stereo photographs of the optic disc.

OBJECTIVE: To evaluate a new automated analysis of optic disc images obtained by spectral-domain optical coherence tomography (SD OCT). Areas of the optic disc, cup, and neural rim in SD OCT images were compared with these areas from stereoscopic photographs to represent the current traditional optic nerve evaluation. The repeatability of measurements by each method was determined and compared. DESIGN: Evaluation of diagnostic technology. PARTICIPANTS: One hundred nineteen healthy eyes, 23 eyes with glaucoma, and 7 glaucoma suspect eyes. METHODS: Optic disc and cup margins were traced from stereoscopic photographs by 3 individuals independently. Optic disc margins and rim widths were determined automatically in SD OCT. A subset of photographs was examined and traced a second time, and duplicate SD OCT images also were analyzed. MAIN OUTCOME MEASURES: Agreement among photograph readers, between duplicate readings, and between SD OCT and photographs were quantified by the intraclass correlation coefficient (ICC), by the root mean square, and by the standard deviation of the differences. RESULTS: Optic disc areas tended to be slightly larger when judged in photographs than by SD OCT, whereas cup areas were similar. Cup and optic disc areas showed good correlation (0.8) between the average photographic reading and SD OCT, but only fair correlation of rim areas (0.4). The SD OCT was highly reproducible (ICC, 0.96-0.99). Each reader also was consistent with himself on duplicate readings of 21 photographs (ICC, 0.80-0.88 for rim area and 0.95-0.98 for all other measurements), but reproducibility was not as good as SD OCT. Measurements derived from SD OCT did not differ from photographic readings more than the readings of photographs by different readers differed from each other. CONCLUSIONS: Designation of the cup and optic disc boundaries by an automated analysis of SD OCT was within the range of variable designations by different readers from color stereoscopic photographs, but use of different landmarks typically made the designation of the optic disc size somewhat smaller in the automated analysis. There was better repeatability among measurements from SD OCT than from among readers of photographs. The repeatability of automated measurement of SD OCT images is promising for use both in diagnosis and in monitoring of progression.

Ability of cirrus HD-OCT optic nerve head parameters to discriminate normal from glaucomatous eyes.

PURPOSE: To determine the ability of optic nerve head (ONH) parameters measured with spectral domain Cirrus HD-OCT (Carl Zeiss Meditec, Inc., Dublin, CA) to discriminate between normal and glaucomatous eyes and to compare them with the discriminating ability of peripapillary retinal nerve fiber layer (RNFL) thickness measurements performed with Cirrus HD-OCT. DESIGN: Evaluation of diagnostic test or technology. PARTICIPANTS: Seventy-three subjects with glaucoma and 146 age-matched normal subjects. METHODS: Peripapillary ONH parameters and RNFL thickness were measured in 1 randomly selected eye of each participant within a 200 × 200 pixel A-scan acquired with Cirrus HD-OCT centered on the ONH. MAIN OUTCOME MEASURES: Optic nerve head topographic parameters, peripapillary RNFL thickness, and area under receiver operating characteristic curves (AUCs). RESULTS: To distinguish normal from glaucomatous eyes, regardless of disease stage, the 6 best parameters (expressed as AUC) were vertical rim thickness (VRT, 0.963), rim area (0.962), RNFL thickness at clock-hour 7 (0.957), RNFL thickness of the inferior quadrant (0.953), vertical cup-to-disc ratio (VCDR, 0.951), and average RNFL thickness (0.950). The AUC for distinguishing between normal eyes and eyes with mild glaucoma was greatest for RNFL thickness of clock-hour 7 (0.918), VRT (0.914), rim area (0.912), RNFL thickness of inferior quadrant (0.895), average RNFL thickness (0.893), and VCDR (0.890). There were no statistically significant differences between AUCs for the best ONH parameters and RNFL thickness measurements (P > 0.05). CONCLUSIONS: Cirrus HD-OCT ONH parameters are able to discriminate between normal eyes and eyes with glaucoma or even mild glaucoma. There is no difference in the ability of ONH parameters and RNFL thickness measurement, as measured with Cirrus OCT, to distinguish between normal and glaucomatous eyes.

Topographic differences in the age-related changes in the retinal nerve fiber layer of normal eyes measured by Stratus optical coherence tomography.

PURPOSE: To determine whether there are regional differences in the age-related changes in peripapillary retinal nerve fiber layer (RNFL) thickness as measured by time-domain optical coherence tomography (OCT). METHODS: Fast peripapillary RNFL scans obtained with the Stratus time-domain OCT with nominal diameter of 3.46-mm centered on the optic disc were carried out on 425 normal participants over a wide age range. One eye was randomly selected for scanning or analysis. Average RNFL-, clock hour-, and quadrant-specific rates of RNFL thickness change were calculated and compared. RESULTS: The 425 study participants ranged in age from 18 to 85 years with mean (±SD) of 46 (±15) years. The mean (±SD) average measured RNFL thickness was 104.7 (±10.8) micrometers (µm). The decline in the average RNFL thickness was 2.4 µm per decade of age. Changes in RNFL thickness per decade of age ranged from -5.4 (P<0.001) at clock hour 1 to -0.9 (P=0.28) at clock hour 6. Similarly, the rate of thickness change per decade of age in the superior quadrant was -4.3 (P<0.001) versus -1.5 (P=0.006) in the inferior quadrant. The slopes of thinning superiorly and inferiorly were highly significantly different (P=0.001). CONCLUSIONS: The age-related decline in normal RNFL measurements does not occur at equal rates around the disc and occurs mainly superiorly.

Normal-tension glaucoma (Low-tension glaucoma).

Glaucoma is now considered an abnormal physiology in the optic nerve head that interacts with the level of intraocular pressure (IOP), with the degree and rate of damage depending on the IOP and presumably the degree of abnormal physiology. Diagnosis of normal-tension glaucoma (NTG), defined as glaucoma without a clearly abnormal IOP, depends on recognizing symptoms and signs associated with optic nerve vulnerability, in addition to absence of other explanations for disc abnormality and visual field loss. Among the findings are a halo or crescent of absence of retinal pigment epithelium around the disc, bilateral pre-chiasmal visual field defects, splinter hemorrhages at the disc margin, vascular dysregulation (low blood pressure, cold hands and feet, migraine headache with aura, and the like), or a family history of glaucoma. Possibly relevant, is a history of hemodynamic crisis, arterial obstructive disease, or sleep apnea. Neurological evaluation with imaging is needed only for atypical cases or ones that progress unexpectedly. Management follows the same principle of other chronic glaucomas, to lower the IOP by a substantial amount, enough to prevent disabling visual loss. However, many NTG cases are non-progressive. Therefore, it may often be wise in mild cases to determine whether the case is progressive and the rate of progression before deciding on how aggressivene to be with therapy. Efforts at neuroprotection and improvement in blood flow have not yet been shown effective.