Journal of Neural Transmission

, Volume 121, Issue 11, pp 1367–1376

Application of an OCT data-based mathematical model of the foveal pit in Parkinson disease

Authors

  • Yin Ding
    • Department of Electrical and Computer EngineeringPolytechnic Institute of New York University
  • Brian Spund
    • North Shore University Hospital
  • Sofya Glazman
    • Department of NeurologySUNY Downstate Medical Center
  • Eric M. Shrier
    • Department of OphthalmologySUNY Downstate Medical Center
    • SUNY Eye Institute
  • Shahnaz Miri
    • Department of NeurologySUNY Downstate Medical Center
  • Ivan Selesnick
    • Department of Electrical and Computer EngineeringPolytechnic Institute of New York University
    • The School of Graduate StudiesSUNY Downstate Medical Center
    • Department of NeurologySUNY Downstate Medical Center
    • Department of OphthalmologySUNY Downstate Medical Center
    • SUNY Eye Institute
    • The School of Graduate StudiesSUNY Downstate Medical Center
Neurology and Preclinical Neurological Studies - Original Article

DOI: 10.1007/s00702-014-1214-2

Cite this article as:
Ding, Y., Spund, B., Glazman, S. et al. J Neural Transm (2014) 121: 1367. doi:10.1007/s00702-014-1214-2

Abstract

Spectral-domain Optical coherence tomography (OCT) has shown remarkable utility in the study of retinal disease and has helped to characterize the fovea in Parkinson disease (PD) patients. We developed a detailed mathematical model based on raw OCT data to allow differentiation of foveae of PD patients from healthy controls. Of the various models we tested, a difference of a Gaussian and a polynomial was found to have “the best fit”. Decision was based on mathematical evaluation of the fit of the model to the data of 45 control eyes versus 50 PD eyes. We compared the model parameters in the two groups using receiver-operating characteristics (ROC). A single parameter discriminated 70 % of PD eyes from controls, while using seven of the eight parameters of the model allowed 76 % to be discriminated. The future clinical utility of mathematical modeling in study of diffuse neurodegenerative conditions that also affect the fovea is discussed.

Keywords

FoveaRetinal imagingOptical coherence tomography (OCT)Parkinson disease (PD)Mathematical modeling

Introduction

Optical coherence tomography (OCT) is very easy to use and has proved remarkably useful in clinical study of the human retina (Huang et al. 1991). OCT is also capable of revealing near-histo-pathologic detail of the retina (Drexler and Fujimoto 2008), a multilayered structure (Bagci et al. 2008). Around the foveola, the inner-retinal layers vanish during normal embryonic development (Provis and Hendrickson 2008; Dubis et al. 2012). The layers are seen to persist in those who have had retinopathy of prematurity (Dubis et al. 2013). In healthy newborn, instead of a flat and homogeneous retinal sheet, a foveal “pit” develops. This pit is readily visible in vivo by slit-lamp biomicroscopy in the human retina, in histological preparations and also evident in OCT images. Individual inner-retinal layers appear to emerge at some distance from the center of the pit (Provis et al. 1998; Tick et al. 2011). The abundance or absence of the inner-retinal cellular elements at various distances possibly determines the appearance and shape of the pit. The neuronal organization of the pit has been studied in postmortem anatomical preparations (Provis and Hendrickson 2008; Hirsch and Curcio 1989). Alterations in the foveal pit shape may allow inferences to be drawn about effect of pathology on the various inner-retinal elements.

Vision is affected in several neurodegenerative diseases, most notably in PD (Bodis-Wollner 2013; Bodis-Wollner et al. 2013) and Alzheimer’s disease (Guo et al. 2010; Leruez et al. 2012; Tzekov and Mullan 2013; Moreno-Ramos et al. 2013). Functional visual impairment in PD was first demonstrated using Visual Evoked Potential measurements (Bodis-Wollner and Yahr 1978) and later retinal function was shown to be abnormal using Electroretinography (ERG) (Stanzione et al. 1990; Moschos et al. 2010). Postmortem studies have revealed altered dopamine (DA) production and morphological impairment in the PD retina (Nguyen-Legros 1988; Ikeda et al. 1994; Djamgoz et al. 1997).

More recently, OCT was used to delineate the structural changes that occur in the foveal retina in PD patients (Altintas et al. 2008; Spund et al. 2013. PD primarily affects the inner-retinal foveal layers (Hajee et al. 2009; Bodis-Wollner et al. 2013), commonly referred to as retinal ganglion-cell complex. DA Amacrine cells are located on the interface between the inner nuclear and inner plexiform layers (Witkovsky 2004). A correlation of inner-retinal thickness as seen on the OCT, and functional visual (contrast sensitivity) changes was shown in a small study in PD (Adam et al. 2013).

Various methods have been proposed for quantitative OCT evaluation. Volumetric measures based on the thickness of the entire (inner and outer) retina have a minimal chance of distinguishing between an attenuated inner retina and a healthy one (Bodis-Wollner et al. 2014). Layer-by-layer analysis through using OCT can detect “remodeling” of the fovea in PD patients (Spund et al. 2013).

The shape of the fovea is somewhat variable even in healthy subjects (Tick et al. 2011). There is also variability between results obtained with different OCT equipment, since they have different data collection methods, algorithms for image formation and derived thickness measurements (Tomlins and Wang 2005; Loduca et al. 2010; Wollstein et al. 2005; Wolf-Schnurrbusch et al. 2009; Sung et al. 2009). The ability to detect and quantify a change in the shape of the foveal pit with a generalizable mathematical model, which can be fitted onto different datasets, may be of clinical value in the study of neurodegenerative conditions.

Mathematical models of the normal foveal pit have used a Gaussian distribution (Dubis et al. 2009) or most recently the double derivative of the Gaussian (Scheibe et al. 2014). The latter study made use of 48 parameters and showed good approximation of the image of the reconstructed fovea to the OCT data-derived foveal shape.

In this paper, we introduce a mathematical description of the foveal pit based on data obtained in age-matched healthy control subjects. Our model is not based on the image created by the OCT equipment. Rather, we quantified the depth of small voxels in a matrix, centered on the foveola and derived the model from the population mean of these finely sampled individual thickness measurements. Thus, our model is not a model of the surface shape of the fovea and neither is the model describing foveal shape riding on the pedestal of the outer retina. Furthermore, our model is not based on interpolating linear demarcation of different layers of the foveal retina. Lastly, the model requires only eight parameters. The healthy control model also describes the PD fovea. With the adjustment of the value of a few selected variables and using receiver-operating characteristics (ROC), we show the power of discrimination of aged-healthy and PD eyes. The results provide the hope that the equation of the fovea can be used for larger sets of PD data obtained on different equipment.

Methods

Study participants

Fifty-four (54) consecutive subjects were enrolled into the study: Parkinson’s Disease (PD) and age-matched Healthy Controls (HC). All study candidates were screened for study eligibility on the basis of general and ocular health and neurological status. Inclusion criteria for the PD patients: ability to give informed consent and comply with study protocol; UK Brain Bank Criteria for PD were used. Neurological exclusionary criteria for the study: young onset (<40 years old) PD; no response to levodopa or dopamine agonist (e.g., ropinirole), cognitive impairment (e.g., memory loss): Mini-Mental State Exam score below 28; Symbol Digit Modalities Test score below 40); CT/MRI of the brain abnormalities suggestive of other cause for parkinsonism (e.g., diffuse white-matter disease or prior stroke); significant drug exposure (e.g., haloperidol); infection preceding parkinsonism; repeated head trauma; other medical conditions such as diabetes mellitus; family history of PD; and history of temporal (cranial) arteritis.

Subjects who were enrolled in the study had thorough neurological evaluation at SUNY Downstate Medical Center and ophthalmic testing and imaging at the SUNY Downstate Long Island College Hospital, Brooklyn, NY. Comprehensive ophthalmic examination consisted of Snellen Visual Acuity and Pelli-Robson Contrast sensitivity and slit-lamp exam of the adnexa, and external and internal media. Intraocular pressure, pupillary reaction, extra-ocular movements and confrontational visual fields were assessed. Optic nerve and macular examinations were conducted with 90-diopter lens at the slit-lamp biomicroscope. Assessment was made for the absence of significant vertical/horizontal asymmetry and notching and normal cup-to-disc ratios (less than 0.5) and absence of retinal nerve-fiber layer defects.

Moderate to highly myopic patients (more than six diopters) and moderate hyperopes (more than four diopters) were excluded from study as significant axial length changes could affect retinal thickness. Peripheral retinal examination was done to rule out retinal vascular disease. Ophthalmic exclusionary criteria included: best-corrected visual acuity worse than 20/40; inability to undergo a dilated fundus exam; intraocular pressure above 22 mmHg; recent cataract surgery or yag-laser capsulotomy (within the past 90 days); diabetes mellitus; any history of a retinal degeneration/dystrophy or vascular disease, optic nerve disease (i.e., optic neuritis or ischemic neuropathy), any form of glaucoma, evidence of prior intraocular inflammation, macular disease, epiretinal membrane or significant macular drusen, and significant hypertensive retinopathy).

Only subjects who met all criteria were then studied. Post hoc OCT criteria for exclusion from the study were unsuspected ophthalmic pathology, such as vitreo-macular traction, cysts, surface-wrinkling, drusen and outer-retinal/photoreceptor disruption. Ultimately, 27 PD patients (50 eyes) and 27 control subjects (45 eyes) fulfilled the stringent criteria. We included OCT measures for each eye, as there is evidence of asymmetry of inner-retinal OCT measurements in PD (Shrier et al. 2012; Cubo et al. 2010). Since one individual’s eye may be more affected in PD, random selection of one of them carries with it the significant risk of missing retinal pathology. Mean age of PD patients and controls was 66.0 ± 8.0 years (range 49–90 years) and 64.6 ± 6.0 (range 53–78 years), respectively. Gender in PD group: 18 males, 9 females; in control group: 10 males, 17 females. Ethnicity of subjects was as follows: in the PD group 16 African descent, 9 Caucasian, and 2 Hispanic; in the control group 19 from African descent, 5 Caucasian, and 3 Hispanic. Mean disease duration in PD patients was 5.8 ± 1.4 years (range 2–10 years). All procedures followed established guidelines, approved by the Institutional Review Board of SUNY Downstate Medical Center.

Equipment, measurements, and statistical analysis

Subjects were examined with Spectral-domain (Fourier) OCT (RTVue RT-100, Optovue, Inc., Fremont, CA) with the MM5 (5 × 5 mm) or EMM5 (6 × 6 mm) scans. Our total data set represented a mixture of both scan types. However, irrespective of this difference, we extracted equivalent data for each eye based on the quantification available through the spectral reflectance values at each pixel obtained in a matrix of 669 × 669 for MM5 scans and 803 × 803 for EMM5 scans, at a radial distance of 1.25 mm from the foveola.

The scans included were all of sufficient quality (signal strength = 75 % of maximal strength) and without any imaging artifacts or distortions. As defined by the makers of the OCT equipment, measures for the “Inner-Retinal Layer” (IRL) thickness include the internal limiting membrane, ganglion-cell layer, and the inner plexiform layer. The “Outer-Retinal Layer” (ORL) includes deeper layers from the inner nuclear layer to the retinal pigment epithelium. The Full Retinal Thickness (FRT) measurements are taken from the internal limiting membrane to the retinal pigment epithelium (IRL–ORL). Thickness measures were taken at a spatial sampling rate of 0.25 mm for the central 2.5 mm foveal diameter (Fig. 1). Care was taken to insure that participants were examined with the measuring grid centered on the anatomic foveola. Raw data files were exported from the Optovue OCT software and analyzed using MATLAB (The MathWorks, Inc. Natick, MA). This allowed us to systematically obtain thickness measurements at adjacent pixels and to manually construct three-dimensional models.
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Fig. 1

OCT measurement grid. The scans included grids of 5 × 5 mm (MM5 scan) or 6 × 6 mm (EMM5 scan) sections of the retina centered on the macula. The yellow dot represents the fixation point. It coincides with the intersection of the vertical and horizontal axis and the center of the measuring grid. Each voxel occupies 0.25 mm on the retina. A cross section of the foveal pit is shown in the vertical and in the horizontal axis. The image is based on the equipment manufacturers algorithm in an aged control subject

Results

The model of the fovea

Our OCT data of retinal thickness at each radial distance from the foveola (up to 2.5 mm for MM5 and 3.0 mm for EMM5) are based on the quantification available through the spectral reflectance values at each pixel obtained in a matrix of 669 × 669 for MM5 scans and 803 × 803 for MM5 scans with a radial distance of 1.25 mm from the foveola for each scan. We exported these values into the MATLAB environment and reconstructed the shape of the fovea for each individual eye and then for the mean of all eyes by group. Based on finely sampled volumetric data of perifoveolar thickness, we developed a mathematical equation to describe the foveal pit as a second-order bivariate polynomial and a zero-mean bivariate Gaussian function as:
$$f(x_{1} ,x_{2} ) = A_{0} + A_{11} x_{1} + A_{21} x_{2} + A_{12} x_{1}^{2} + A_{22} x_{2}^{2} + K\;\exp \left( { - \frac{{x_{1}^{2} }}{{2\sigma_{1}^{2} }} - \frac{{x_{2}^{2} }}{{2\sigma_{2}^{2} }}} \right)$$
(1)
The model consists of two components: a second-order bivariate polynomial and a zero-mean bivariate Gaussian function (Fig. 2). The polynomial component is intended to capture the coarse behavior; the Gaussian component is intended to capture the dip at a finer scale. In the model, there are a total of eight parameters: five for the polynomial component and three for the Gaussian component. In equation (1), the variables ×1 and ×2 are commutative as the axes or directions from nasal to temporal, and from inferior to superior. They refer to the inner retina (inner plexiform and ganglion-cell layer). Here, we fixed the variable ×1 is the direction from temporal to nasal covering the space from −1.25 to 1.25 mm, and ×2 is from superior to inferior, with (×1, ×2) = (0, 0) is exactly at the foveola, therefore, a positive number of A11 and A21 mean an increasing slant from temporal to nasal, and from inferior to superior, respectively. While the parameters A11 and A21 give the first-order trend of the thickness, A12 and A22 give a finer quadratic curvature fitting of two directions, respectively. Note that we model the shape as subtracting a crater (with the maximum height to be zero) describing as a bivariate Gaussian, from a bivariate polynomial surface, the parameter A0 should give the rough thickness of the central area with a “filled” crater, where K gives the depth of the crater and σ2 gives the width.
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Fig. 2

The two surfaces shown here are the polynomial and Gaussian components of the model, which is based on the difference of the two components (see text for further explanation)

A numerical optimization algorithm for the foveal model in PD and healthy eyes

We also note that we do not constrain the Gaussian function in the model to have unit area, because the data are not normalized in this way and because the tails of the Gaussian extend beyond the window of the model. We also initially evaluated a difference of two Gaussians for the foveal pit, but the wider of the two Gaussians is not well determined numerically because only the middle portion of the wider Gaussian is covering the central fovea we are analyzing. The variance σ2 of the wider of the two Gaussians is not well determined from the data and the optimization of the parameters is not well conditioned. To obtain the parameters of the model from data, we use an optimization approach. Specifically, we minimized the volume between the model and the mean data of healthy controls (Fig. 3). The minimization problem can be written as:
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Fig. 3

The inner-retinal thickness map of the fovea in controls, based on small voxel volumetric measures, compared to their model. Here, color coding does not represent thickness, but rather a difference function. Red represents a large difference while blue represents minimal difference. The maximum difference (error) is roughly 4 microns between the data and the model

$$\arg {\kern 1pt} \mathop {\hbox{min} }\limits_{p} \int\limits_{{x_{1} }} {\int\limits_{{x_{2} }} {\left| {f(x_{1} ,x_{2} ) - } \right.\left. {d(x_{1} ,x_{2} )} \right|dx_{1} x_{2} ,} }$$
(2)
where p is the parameter set, p = {A0, A11, A12, A21, A22, K, σ1, σ2}. Here, d(×1, ×2) represents the OCT scan data of the retinal area of interest. More specifically, during the implementation, the data are in a 2D array with a size of 335 by 335 truncated from MM5 or EMM5 raw data centered on the foveola as the origin. F (×1, ×2) depends on the seven parameters. Because no closed form solution exists to this minimization problem, we use a numerical optimization algorithm. The computation is performed using MATLAB (Version 7.8) with the function fmin to search, which uses the Nelder-Mead simplex optimization algorithm. The three-dimensional representations of the foveal model (both for PD patients and healthy controls) are shown in Fig. 4a–d, respectively. When we compared the model parameters for controls and PD patients, we found that the difference was the largest in the term A11 (Table 1).
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Fig. 4

Reconstructed foveae using MATLAB tools. In all these images the fovea is seen from an angle in 3 dimensions on the left and from above on the right. Color coding represents true thickness values derived from voxel depth measurements Thickness is measured from the normalized reference of the foveola to the inner limiting membrane a the fovea in healthy control subjects, b in PD subjects. Note a difference in the average redness (thickness of the fovea between a and b. c The model derived from the healthy control data. d The model applied to the average PD fovea. Note that the model retains differences seen in the “raw” data between healthy controls and PD subjects. These differences are: reduced overall thickness of the PD fovea and secondly one may discern a difference in the blue pit diameter. This difference is better demonstrated in the rigorous comparison of the model parameters between healthy controls and PD subjects. (see text)

Table 1

The columns represent the variables of the foveal model and their numerical values in controls and PD patients

PD/Control

#

A0

A11

A12

A21

A22

K

σ1

σ2

PD

45

90.65

0.84

−5.S3

−0.0036

−5.91

−89.38

0.42

0.36

Control

50

104.93

3.91

−8.74

−0.5872

−8.38

−103.25

0.45

0.37

Note that A11 shows about fourfold change between control and PD values. The numerical value of A22 is vanishing suggesting that it contributes little power for discrimination

The ROC curve of the parameter A11 in the foveal slope

The ROC curve was used to explore the performance with the perspective of probability (Hanley and McNeil 1982) where only one condition is adopted for classification (in this case A11 only). We adapted the model to all the individual data, so that a set of parameter p can be extracted as a feature for each eye. We were most interested in the single parameter A11 because it gives the foveal slant trend from temporal to nasal. We also analyzed the effect of all variables (see below). The ROC curve of A11 is shown in Fig. 5, and the value of the Area Under Curve (AUC) is 70.13 %, which is the accuracy of only using A11 extracted from the model as a classifier.
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Fig. 5

Receiver Operating Characteristics (ROC) was applied to one of the variables of the mathematical model of the fovea in controls and PD subjects. The curve is based on true and false positive values of belonging to the PD classification. The Area under the curve (AUC) value is around 0.7. This single variable thus identifies 70 % of PD subjects and misidentifies 30 % of healthy aged control subjects. Longitudinal and larger scale studies may affect this yield

The optimal combination of the foveal model parameters in PD

To evaluate the efficiency of various parameter combinations for the purpose of classifying eyes as either control or PD, we use logistic regression (with the criteria of minimizing classification error) (Bishop 2006). Identifying subsets of parameters with high discrimination power may also be useful in future work. We have considered each of the 255 combinations of the 8 parameters in the parameter set, p. For each combination of parameters, we generate the true positive, false negative and total error rate of classification. In MATLAB implementation, we used the build-in function mnrfit for this method (Matlab Statistics Toolbox Release 2011). Note that, when a single parameter is used for logistic regression, the true positive and false negative correspond to a point on the ROC curve. As a result, we made a table for “best combinations” including different numbers of parameters (Table 2). Using all the eight parameters gives a true positive rate of 72 %, false positive rate of 33.33 %, and 30.53 % error rate. However, application of seven parameters (except A0) has the best diagnostic value, with 76 % true positive rate, 28.89 % false positive rate and 26.32 % error rate.
Table 2

The optimal combination of the foveal model parameters in PD, using logistic regression

M

Parameter

Error rate (%)

True positive (%)

False positive (%)

1

A11

33.68

74.00

42.22

2

A11σ2

30.53

76.00

37.78

3

A11A22K

29.47

68.00

26.67

4

A11A12A22σ1

28.42

78.00

35.56

5

A11A12A21A22σ1

27.37

80.00

35.56

6

A11A12A21A22σ1σ2

27.37

78.00

33.33

7

A11A12A21A22K σ1σ2

26.32

76.00

28.89

8

A0A11A12A21A22K σ1σ2

30.53

72.00

33.33

The best discriminating power is achieved using all parameters except A0, with 76 % true positive rate, 28.89 % false positive rate, and 26.32 % error rate. The worst result is for using σ2 alone: true positive = 80 %, false positive = 93.33 %, error rate = 54.74 %

Discussion

OCT and the mathematical model for foveal pit morphology

Using the clinically well-established technique of slit-lamp ophthalmoscopy, one is capable of detecting gross retinal pathology which is apparent on the surface of the retina or that distorts the foveal pit. Individual layers remain invisible to the ophthalmoscope. OCT, however, allows a penetrating “look” into all retinal layers (Bodis-Wollner et al. 2014), which includes the superficial NFL and the retinal GC-s. The NFL is comprised of axons of retinal ganglion cells. NFL is affected in diseases of the optic nerve, for instance in glaucoma (Budenz et al. 2008). Thinning of the NFL and even deeper retinal layers has been reported in different neurodegenerative disorders such as Alzheimer (AD) (Guo et al. 2010; Brandt et al. 2011; Moreno-Ramos et al. 2013; Kirbas et al. 2013) and PD (Inzelberg et al. 2004).

A meaningful comparison of various studies is difficult without specifying the particular region of interest. The very small area of the foveal pit is important for our best vision and macular pathology is appreciated to be of major public health concern as it is subject to the rigors of aging. Pathology affecting the very small area of the fovea, as it occurs in diabetes and other maculopathies, is not well seen on ophthalmoscopy until the pathology is clinically advanced and distorts the normal architecture of the pit (Sikorski et al. 2013).

Retinal thickness depends on the distance from the foveola. In the foveolar center, there are only photoreceptors, which henceforth make up the entire volume here (Provis and Hendrickson 2008). With increasing distance away from the foveola, there are more and more neural elements, except within the central 1 mm where there are few, if any, retinal ganglion cells. With distance of 1–1.2 mm more and more and diverse neural elements are included in a macular volume measurement until finally the thickness of the ganglion cells and nerve fibers contributes to the average volume (Provis and Hendrickson 2008). Some algorithms for detecting maculopathy quantify retinal volume in successively larger volumes surrounding the foveola (Forooghian et al. 2008; Sakata et al. 2009). This macular volume averaging may wash out any potential differences that are caused by diverse pathologies potentially affecting different types of neurons. Say for instance the total retinal thickness is 300 micron at a given distance from the foveola while the inner complex is thinned by 12 microns. To detect this small amount of thinning based on total retina thickness would require a very large number of measurements.

In study of diabetic retinopathy, perimacular retinal tissue volumes are usually averaged, blurring potential differences between specific retinal layers and obfuscating cellular loss (Bodis-Wollner et al. 2014). In averaging of retinal volumes detailed information is lost regarding the contribution of inner-retinal horizontal interconnections. This would be critical in study of neurodegenerative disease such as PD, given that the deficit described is in the inner plexiform layer.

Based on the foregoing considerations we created a model of the fovea such that the model relies on the continuous interconnected neuronal properties. Foveal models were previously developed by Dubis et al. (2009) and Scheibe et al. (2014) for healthy human fovea. Dubis et al. (2009) developed a foveal pit mathematical model according to OCT thickness data in healthy subjects. Subjects they included in their study were 39 emmetropic and 22 myopic patients, with mean age of 26.3 years. They extracted three main metrics from their model, including pit depth, diameter, and slope. In applying the model to individuals’ OCT data they then suggested these metrics would be useful to detect retinal changes in diseases including retinopathy of prematurity, high myopia, and normal prenatal foveal development. Recently, another mathematical model, also based on a Gaussian was developed by Scheibe et al. (2014) for healthy subjects. It is based on fit to radial lines of the fovea, a four-parameter one-sided generalization of the second derivative of the one-dimension Gaussian function. Twelve radial directions were used in their work, which leads to a total of 48 parameters. As a result, an accurate reconstruction of the foveal shape is possible, but it is complicated by the high number of model parameters.

Our model is based on the difference of a Gaussian and a polynomial which allows a spatial definition (in the retina) without interpolation and without extrapolation based on the Gaussian characteristics. The importance of not using interpolation to derive linear demarcation of retinal layers is that a linear interpolation over extended retinal distances is bound to have assumptive statistical errors. The use of a difference of a Gaussian and a polynomial allows the user to set the spatial extent of the model and adopt it to the region (s) of interest.

Ultimately the model we developed was the one with the least possible residual error to the results of the actual population mean (see “Methods and Results”). Our equation specifies the vertical and horizontal slopes of the foveal pit, its thickness and depth and the overall shape of the fovea. The parameters of the equation relate to the height, horizontal and vertical spread of each component and the nasal-temporal and superior-inferior slopes of each of the components of the model. Consistent with the histological images of the foveal pit, the model and the reconstructed images show that the slope of the foveal pit is not adequately represented with a linear approximation as determining the highest point. The slope is curvilinear. The equation was fitted to the control and PD data. PD did not change curvilinearity; there is a change only in selected coefficients as compared to healthy controls. We show that empirically, combining seven of the eight variables of our model is capable of discriminating over 76 % of patients.

Demographic variables and foveal pit morphology

Several studies reported a race- and sex-related difference in foveal pit morphology (Wagner-Schuman et al. 2011; Song et al. 2010; Kashani et al. 2010). The central subfield thickness, average retina thickness and overall macular volume are shown to be reduced in women (Wagner-Schuman et al. 2011; Kashani et al. 2010; Song et al. 2010). These differences can lead to a sex-related difference in appearance of the foveal pit. Furthermore, race is another factor potentially affecting the width of the pit. American–African race is introduced as a predictor of decreased mean foveal thickness (Kashani et al. 2010). Despite the population demographic variability, our model is considering all different parameters in the foveal pit morphology, and it successfully fits to eyes of both gender, and different races. Further studies may be needed to evaluate though if individual variables of the model are sensitive to these demographic differences. Axial length is another factor which potentially could inversely affect macular thickness and macular volume (Song et al. 2010). However, we excluded all moderate or highly myopic and hyperopic patients from our study to omit the effect of extreme axial length differences on the foveal model.

The potential clinical relevance of the model in Parkinson disease

Over the last decade a search for easily available, inexpensive, non-invasive markers for the diagnosis, quantification of progression and response to therapy in neurodegenerative disease such as PD has markedly accelerated. Besides the brain, current interest also focuses on imaging of the retina. OCT promises to be useful in charting the progression and potential response to therapy in PD and some other neurodegenerative diseases (Spund et al. 2013; Guo et al. 2010; Altintas et al. 2008; Pulicken et al. 2007). Some of the neurodegenerative diseases affect vision and cause impaired retinal signal processing (Bodis-Wollner 2013; Parisi et al. 2001). OCT studies revealed retinal thinning and remodeling of the fovea in PD (Spund et al. 2013; Bodis-Wollner et al. 2014). OCT is able to hone in on the foveal region and the peripapillary region (Bodis-Wollner et al. 2014). In PD the region of interest is ganglion-cell complex (Bodis-Wollner et al. 2014) and the nerve-fiber layer. In Alzheimer’s disease, defects are found in the peripapillary nerve-fiber layer (PPNL) (Fortune et al. 2013; Young et al. 2013; Kromer et al. 2014). We developed a mathematical model based on the shape and volume, symmetry and asymmetry, size and thickness of the fovea. Our model for describing three-dimensional foveal shape change in PD based on interconnected, but discrete pixel volumes gives a manageable set of results, without loss of relevant information. Our results show that in PD, seven of the eight variables of the mathematical based are sufficient to discriminate PD from age-matched healthy subjects. We propose that this model be used in a larger cohort of PD patients and on different OCT instrumentation.

Acknowledgments

Samantha Slotnick, OD, for many critical discussions and comments on the study; Dr. Mike Sinai for his essential guidance in obtaining the data structure from the OCT equipment and many good suggestions; the late Dr. Wayne March for actively participating in the eye evaluation of the subjects; Liu Tong for contribution for the imaging of the foveal pit; Muhammad Javaid, MD, Aleksander Belakovskiy, Galina Glazman, and Zoya Belakovskaya for help in data collection; Jerome Sherman, OD, for many useful clinical discussions; Douglas Lazzaro, MD, for general support. The Michael J. Fox Foundation for financial support and the Research to Prevent Blindness for departmental research support.

Copyright information

© Springer-Verlag Wien 2014