Preliminary diagnosis of eye health via wavefront parameters based on statistical frequency distributions

Abstract

Background

Eye disease is a global health issue, and current diagnostic methods are complex, costly and more suitable for final diagnosis. In order to detect abnormal eyes earlier, it is necessary to develop a simple and low-cost method for preliminary diagnosis of eye health. Ocular aberration measurement is a safe routine eye examination, and using its results for diagnosis will be valuable.

Methods

Using the wavefront parameters of ocular aberrations for diagnosis, including root-mean-square (RMS), peak-to-valley and entropy, were presented in this work. A combined diagnosis was also made with their results through majority voting. The frequency distributions of the parameters for normal and abnormal eye groups were statistically analyzed from 610 sets of aberrations. Subject was diagnosed as the group with a higher frequency by comparing its parameter with the distributions, because it has a higher probability of belonging to the dominant group. 66 test samples were diagnosed, and diagnostic performance was calculated by comparing their results with pathological decisions. The performance of using different parameters and order aberrations for diagnosis was investigated.

Results

For the two groups of each parameter by using the 2nd−8th orders, the data distributions are significantly different at significance level of 0.05 via both Mann-Whitney test and t-test, and the frequency distributions are also obviously different. The method by using the RMS parameter and the 2nd−8th orders has the best performance, with a sensitivity of 81.8%, a specificity of 90.9%, an accuracy of 86.4%, a positive predictive value of 90.0%, and a negative predictive value of 83.3%.

Conclusions

The best performing method can basically meet the preliminary diagnostic requirement of eye health. It is safe, simple, low-cost, and potentially applicable in abnormal eye screening and assisted medical shunting.

Supplementary Information

The online version contains supplementary material available at 10.1186/s40001-026-03839-6.

Introduction

Eye health has widespread and profound implications for life, health, development, economy, etc. [1], and eye disease is a major global health issue. According to a report in the journal “The Lancet Global Health”, an estimated 596 million people experienced distance vision impairment, 43 million of whom were blind; and another 510 million people experienced near vision impairment worldwide in 2020 [1]. Eye diseases are also remarkably common and affect every individual. A report released by the World Health Organization in 2019 stated that those who live long enough would experience at least one eye condition in their lifetime [2]. Hence, attention should be given to regular and irregular monitoring of eye health to detect abnormal eyes in time. At present, eye disease diagnosis is performed mainly with the help of various advanced instruments, such as fundus camera [3, 4], scanning laser ophthalmoscope [5], optical coherence tomography [6, 7], fundus fluorescein angiography [8], indocyanine green angiography [9], and optical coherence tomography angiography [10–12]. These methods have high diagnostic accuracy, but can be performed only by professional medical doctors and impose an economic burden on patients. Thus, they are more suitable for final diagnosis. Developing a simple and low-cost method for the routine examination and preliminary diagnosis of eye health is highly valuable. Here, the eye health is limited to denote the medical normal or abnormal state, i.e., without or with eye condition.

Rapid, objective and completely automated measurement of ocular aberrations has been achieved with the advancement of wavefront-sensing technologies [13], including Shack-Hartmann aberrometry [14, 15], laser ray-tracing aberrometry [16, 17], Tscherning aberrometry [18, 19], spatially resolved refractometry [20, 21], etc. Accordingly, aberration measurement has become a necessary procedure for vision correction, lesion examination and disease diagnosis. As the first procedure, if its results can be used to make a preliminary diagnosis of eye health, it can be avoided that all patients go to the hospitals to consume limited medical resources and doctors' time. The diagnostic result can provide a reference for subsequent treatments, i.e., play a medical shunting role. People with eye conditions can be screened through aberration measurement via physical examination, and they can perform self-management of eye health or seek medical treatment as soon as possible. In addition, a promising trend of aberrometer is to enter home for daily optometry with its simple, miniature and low-cost advancement. If it also has an additional diagnostic function, making it a family healthcare product, it will be of great significance for eye disease prevention and will also have economic value.

Several studies [22–24] have shown that there are differences in the statistical distributions of ocular aberrations between normal and abnormal eyes, especially high-order aberrations (HOAs). Bessho et al. [22] studied the aberrations of 82 eyes from 66 patients with macular disease and 85 eyes from 51 patients without disease, and the result showed that the HOAs of the diseased eyes were significantly greater than those of the normal eyes. Xiao et al. [23] analyzed the aberrations in Chinese eyes, including 33 eyes from 22 patients with glaucoma, 107 eyes from 68 patients with diabetic retinopathy, and 186 eyes from 103 patients without pathological features. The results revealed that the HOAs of the glaucoma and diabetic retinopathy eyes were ~2.9 and ~1.8 folds greater than those of the normal eyes, respectively. Xiao [24] also statistically analyzed 676 sets of aberrations from 332 healthy eyes and 344 diseased eyes, and the result also showed that the HOAs of the diseased eyes were greater than those of the healthy eyes. These differences may provide the possibility to use ocular aberrations for differentiating normal and abnormal eyes.

At present, ocular aberrations are mainly used as auxiliary information for lesion examination, disease diagnosis, guidance of refractive surgery [25–27], and evaluation of surgical outcomes [28] in clinical applications. But they have never been solely used for diagnosis. Aberrations are commonly expressed by Zernike polynomials and can be quantitatively described by Zernike coefficients [29]. Therefore, we previously used the coefficients for diagnosis based on their statistical frequency distributions through majority voting. Firstly, the frequency distributions of the coefficients for normal and abnormal eye groups were obtained. Then, the coefficients of a subject were determined belonging to normal, abnormal or undetermined by comparing them with the distributions. Finally, the diagnostic result was made according to the numbers of the coefficients determined as normal and abnormal. Using the coefficients of the 2nd, 3rd, 7th and 8th orders, as well as 33 coefficients after excluding the 9 coefficients with close distributions, respectively, have the best diagnostic performance. They have high specificity and positive predictive value, moderate accuracy and negative predictive value, but very low sensitivity (the highest is only 66.7%). Low sensitivity limits the ability to detect abnormal eyes. Hence, the overall performance was unsatisfactory, and the previous work was unpublished.

In contrast to the above, which took the way of aberration decomposition (used the individual coefficients for diagnosis), this work took the way of aberration combination (used the entire wavefront for diagnosis). Here, we propose to use wavefront parameters for the preliminary diagnosis of eye health, including root-mean-square (RMS), peak-to-valley (PV) and entropy (ENT) parameters. The frequency distributions of each parameter for the normal and abnormal eye groups were statistically analyzed from 610 sets of aberrations. A subject was diagnosed by comparing its parameters with their corresponding distributions, respectively. A combined diagnosis of the three parameters was also made with their diagnostic results through majority voting. The objectives of this work are to investigate whether the parameters can be used for the preliminary diagnosis of eye health based on their frequency distributions, and which parameter and order aberrations have the best performance.

Methods

Aberration data collection

A total of 676 sets of ocular aberrations [24] were clinically collected from Chinese eyes at the Eye Hospital of Wenzhou Medical University and the West China Hospital of Sichuan University. They were measured from 439 adult patients with commercial aberrometer (Topcon KR-1W, Tokyo, Japan) at a pupil diameter of 6 mm and 820 nm wavelength. The aberrations of each eye were quickly measured three times via the automated adjustment and acquisition mode of the aberrometer, and the averages of the three measurement results were used as the final results [24]. The eye health of the patients was pathologically diagnosed by ophthalmologists, and the aberration data were accordingly divided into normal and abnormal eye groups. The former contains 332 eyes (170 left and 162 right) from 204 patients without pathologic features and/or only with refractive error. The latter contains 344 eyes (163 left and 181 right) from 235 patients with various diseases, including amblyopia (n=34, 9.9%), maculopathy (n=45, 13.1%), retinopathy (n=51, 14.8%), glaucoma (n=72, 20.9%), diabetic retinopathy (n=110, 32.0%) and other diseases (n=32, 9.3%). The samples were from 142 males and 190 females for the normal group, and from 158 males and 186 females for the abnormal group.

Aberrations were expressed by Zernike polynomials, and their arrangement was based on the standard recommended by the Optical Society of America [29]. The measured aberrations were saved in an Excel table in the form of Zernike coefficients C₀ to C₄₄, which contain the first eight orders and can meet common application requirements. However, the zero order (piston) cannot be measured accurately at all [30]. It and the first order (tip and tilt) do not supply any information about the characteristics of the eye itself [30] and also not affect the visual or imaging quality [31]. Hence, they were excluded. Only the aberrations from the 2nd to 8th orders, corresponding to coefficients C₃ to C₄₄, were used for analysis. About 10% of the samples, i.e., 33 sets of data, were randomly taken from each group to form test groups. The remaining 299 sets of normal data and 311 sets of abnormal data were used to form statistical groups. The grouping ratio of 1:9 is merely a common empirical value, which purpose is to use more samples for statistical analysis to make the frequency distributions more accurate.

Wavefront parameters of aberrations

The wavefront $W(\rho ,\theta )$ in polar coordinates is reconstructed by

$$W(\rho ,\theta )=\sum C_k·Z_k(\rho ,\theta )$$ 1

where $\rho$ and $\theta$ are the radial and azimuthal coordinates, and are in the ranges of 0−1 and 0−2π, respectively; $Z_k(\rho ,\theta )$ and C_k are the k-th Zernike polynomial and coefficient, respectively; and the index k is in the range of 3−44. To be convenient for subsequent processing, the $W(\rho ,\theta )$ is converted to the wavefront $W(x,y)$ in Cartesian coordinates.

The RMS and PV are the most often used optical metrics of the wavefront over the entire pupil. The RMS is a simple criterion of the wavefront, which packs complex aberrations into a single number [30]. The PV represents the dynamic range of the wavefront, which depends heavily on the two extreme values [30]. It does not necessarily consider wavefront quality well [32], but is still adopted in this work to form an odd number of metrics to make a combined diagnosis of multiple parameters through majority voting. They are expressed as:

$$RMS=\sqrt{\frac{1}{M·N}{\sum }_{j=1}^N{\sum }_{i=1}^M{(W(i,j))}^2}$$ 2

$$PV=max(W(i,j))-min(W(i,j))$$ 3

where M×N are the numbers of data points $(i,j)$ and are 800×800 points in this work; max and min are the maximum and minimum values, respectively.

The ENT was inspired by an information theory approach to optics [33, 34]. It is a statistical measure of randomness, and usually used to measure the complexity of intensity image and the amount of information contained. Currently, it has been successfully applied in various fields. This powerful method may help distinguish between the normal and abnormal eyes, and thus be used as a diagnostic parameter. It is defined as [35]:

$$ENT=-\sum \left[p·log_2^{}(p)\right]$$ 4

where p is the histogram count of the image. The wavefront W(i, j) has positive and negative values and needs to be mapped to a wavefront image WI(i, j) for ENT calculation. To obtain the largest dynamic range of the image, the maximal value range of W(i, j) for all the data (including the statistical and test groups) is mapped to the maximal intensity range of 0−255 in the WI(i, j) with an 8-bit gray level. The wavefront images by using different orders in this work are also generated in this way.

Data processing was performed with Matlab software (R2007b Version, MathWorks, Inc., Natick, USA). They can be directly calculated via the existing function commands as: RMS=sqrt(mean(W(i, j).^2)), PV=max(W(i, j))-min(W(i, j)), and ENT=entropy(WI(i, j)), where sqrt( ), mean( ), max( ), min( ) and entropy( ) are the commands to calculate the square root, average, maximum, minimum and entropy values of the input within the brackets. Taking the #1 normal and #1 abnormal test samples as examples, their wavefronts and wavefront images are shown in Fig. 1. Their parameter values are also given in the subfigures. Note that only the data within the circular areas are valid and processed.

Fig. 1.

Wavefronts (a and b) for RMS and PV calculations and wavefront images (c and d) for ENT calculation, generated with 2nd−8th order aberrations. #1 NTS and #1 ATS: #1 normal and #1 abnormal test samples, respectively

Statistical frequency of wavefront parameters

The RMSs and PVs of the wavefronts generated with the aberrations from the statistical groups, and the ENTs of their corresponding wavefront images were obtained with the above methods. Then they were statistically analyzed to obtain the frequency histograms of the normal and abnormal eye groups. The bin size and the distribution range of each parameter must be the same for the two groups, and all the bin sizes in this work were in the range of 52−57. Since the total sample numbers were different for the two groups, the relative frequency (RF) was used, which is the ratio of the sample number within a bin to the total sample number. The RF distributions of the parameters were plotted by using the median values in the bins as the abscissa and the RFs as the ordinate. Taking the RMS as an example, its RF distributions are shown in Fig. 2. All the statistical analysis was performed with Origin software (2016 Version, OriginLab Corp., Northampton, MA, USA).

Fig. 2.

Diagnosis of subject by taking RMS parameter as an example. RMS₁−RMS₅: RMS values of subject; N, A and U: normal, abnormal and undetermined, respectively

Diagnostic method

A subject is diagnosed by comparing its parameter with the frequency distributions, based on the fact that it has a higher probability of belonging to the dominant group on the distributions corresponding to its parameter. The RMS parameter is taken as an example to describe the diagnostic method, as shown in Fig. 2. When the RMS value of a subject is compared with its distributions, there are five possible situations and the subject is accordingly determined as the following three results. When the RF of the normal group corresponding to the RMS value (e.g., RMS₂) is greater than that of the abnormal group, the subject is diagnosed as normal (marked as N). When the RF of the abnormal group corresponding to the RMS value (e.g., RMS₁) is greater than that of the normal group, it is diagnosed as abnormal (marked as A). When the RFs of both groups corresponding to the RMS value (e.g., RMS₃) are equal, or those corresponding to the RMS value (e.g., RMS₄) are zero, or the RMS value (e.g., RMS₅) is not within the distribution range, it is undetermined (marked as U).

For instance, the #1 normal and #1 abnormal test samples have the RMS values of 5.78 and 2.97 μm, respectively. The RF of the normal group corresponding to the former is greater than that of the abnormal, whereas the RF of the abnormal group corresponding to the latter is greater than that of the normal. Hence, they are diagnosed as normal and abnormal, respectively. All the test samples were diagnosed in this way to obtain their diagnostic results by using each parameter. A combined diagnosis is finally made with the three results through majority voting: the result with a larger number is considered as the combined diagnostic result.

Performance evaluation

The pathological decisions of the test samples are used as the standard to judge whether their diagnostic results are correct. The following four results can be obtained: true positive (TP, pathological positive sample is diagnosed as abnormal with the method), false positive (FP, negative is misdiagnosed as abnormal), true negative (TN, negative is diagnosed as normal), and false negative (FN, positive is misdiagnosed as normal). The samples diagnosed as undetermined do not participate in the performance evaluation. The numbers of the TP, FP, TN and FN cases are counted, and then diagnostic performance can be evaluated by using the five common metrics as: Sen=TP/(TP+FN), Spe=TN/(FP+TN), Acc=(TP+TN)/(TP+FN+FP+TN), PPV=TP/(TP+FP), and NPV=TN/(FN+TN), where Sen, Spe, Acc, PPV and NPV are the sensitivity, specificity, accuracy, positive and negative predictive values, respectively. Diagnostic yield (DY) is also calculated to quantify undetermined cases as: DY=(TC−UC)/TC, where TC and UC are the numbers of the total samples and undetermined cases, respectively.Due to its high values (96.7−100% for all the 24 occasions involved in this work), the DY is not a crucial metric and does not need to be presented.

Results

Frequency distributions of wavefront parameters

Fig. 3 shows the RF distributions of the parameters using the 2nd−8th orders. The following findings can be obtained by analyzing them. (1) All the parameters do not present any standard distribution law, e.g., Gaussian distribution. The overall trends of the RMS and PV are consistent; and the fluctuations in the PV, especially in the ENT, are more intense; (2) For all the parameters, the values corresponding to the high RF regions for the normal group are larger than those for the abnormal group. The RF peaks and their corresponding parameter values for the normal and abnormal groups, respectively, are as follows: 9.0% @ 5.40 μm and 15.8% @ 1.40 μm for the RMS, 8.7% @ 22.00 μm and 13.8% @ 7.00 μm for the PV, and 9.7% @ 6.25 dB and 5.1% @ 4.05 dB for the ENT; (3) For the abnormal group, the RMS and PV are highly concentrated in the small value region, whereas the concentrated distribution of the ENT is not obvious. For the normal group, the RMS and PV are concentrated in the region slightly left to the central, whereas the ENT is highly concentrated in the large value region; and (4) The regions dominated by the two groups, respectively, are divided by two lines schematically. For the RMS, the number of the normal eyes in the region of 3.83−11.16 μm shown by the two lines accounts for ~78.6% of the total, whereas the number of the abnormal eyes in the region of ≤3.83 μm accounts for ~70.1%. For the PV, the number of the normal eyes in the region of 16.54−41.04 μm accounts for ~73.6%, whereas the number of the abnormal eyes in the region of ≤16.54 μm accounts for ~64.6%. For the ENT, the number of the normal eyes in the region of 5.53−7.28 dB accounts for ~79.6%, whereas the number of the abnormal eyes in the region of ≤5.53 dB accounts for ~70.1%. Therefore, there are obvious differences in the distributions of each parameter between the two groups, making it possible to diagnose the eye health of a subject based on them.

Fig. 3.

RF distributions of wavefront parameters by using 2nd−8th order aberrations. a RMS, b PV, and c ENT

Diagnostic results of test samples

By comparing the parameters of a test sample with their corresponding RF distributions shown in Fig. 3, its diagnostic results by using different parameters can be obtained. Still taking #1 normal and #1 abnormal test samples as examples: the former was diagnosed as normal by the RMS, PV and ENT, respectively; and the latter was diagnosed as abnormal by the RMS and ENT, respectively, but as normal by the PV (an FN case). A combined diagnosis was made with above the results, and they were diagnosed as normal (3 normal vs. 0 abnormal) and abnormal (1 normal vs. 2 abnormal), respectively. For simplicity, the above methods are referred to as the RMS, PV, ENT and Comb methods, respectively. All the test samples were treated similarly, and their diagnostic results by using the four methods, respectively, are listed in Table S1 in Supplementary material 2.

Diagnostic performance

The numbers of the samples diagnosed as normal, abnormal and undetermined by these methods, respectively, are listed in the last row in Table S1. Taking the results obtained with the MRS method as examples: among the 33 normal samples, 30 and 3 samples were diagnosed as normal and abnormal, respectively, briefly written as 30N3A; among the 33 abnormal samples, 6 and 27 samples were diagnosed as normal and abnormal, respectively. The numbers of the TP, FP, TN and FN cases could be obtained by comparing the diagnosed results with their pathological decisions, and then its diagnostic performance was calculated and listed in Table 1. The RMS method has a sensitivity of 81.8%, a specificity of 90.9%, an accuracy of 86.4%, a PPV of 90.0% and an NPV of 83.3%. Similarly, the performances of the PV, ENT and Comb methods were also obtained and listed in Table S2 in Supplementary material 2.

Table 1.

Pathological decisions versus diagnostic results by using RMS method and 2nd−8th order aberrations

		Pathology
		Positive	Negative	Total
Diagnosis	Abnormal	27 (TP)	3 (FP)	30	PPV=90.0%
	Normal	6 (FN)	30 (TN)	36	NPV=83.3%
	Undetermined	0	0	0	DY=100.0%
	Total	33	33	66
		Sen=81.8%	Spe=90.9%	Acc=86.4%

All the metrics are summarized in Table 2 for comparison. A simple method was used to qualitatively evaluate their performance. The best and second-best values of each metric were first marked in bold and italics, respectively. The numbers of the best and second-best metrics of each method were then counted and listed in the last column. For example, the RMS method has five best and zero second-best metrics, briefly written as 5B0SB. The overall performance of the methods is finally evaluated according to the numbers of the best and second-best metrics. The RMS method has all the best metrics and thus has the best performance; and the Comb and PV methods have the second-best and worst performance, respectively.

Table 2.

Diagnostic performance of different methods by using 2nd−8th order aberrations (units: %)

Method	Sen	Spe	Acc	PPV	NPV	Evaluation
RMS	81.8	90.9	86.4	90.0	83.3	5B0SB
PV	72.7	75.8	74.2	75.0	73.5	0B0SB
ENT	72.7	90.9	81.8	88.9	76.9	1B0SB
Comb	78.8	90.9	84.8	89.7	81.1	1B4SB

B and SB: the best in bold and the second-best in italics, respectively

Diagnostic performance using other order aberrations

Some studies [22–24] have shown that there are differences in the HOAs (from the 3rd order to up) between normal and abnormal eyes. It is easy to think that better results may be obtained if only using the HOAs for diagnosis. Hence, the performance by using the 3rd−8th orders for diagnosis was studied. All the data processing was the same as for the 2nd−8th orders, and the obtained RF distributions are shown in Fig. S1 in Supplementary material 1. For each parameter, the distributions of the two groups are relatively close and not clearly distinguished from each other. For all the parameters, the values corresponding to the high RF regions for the normal group are slightly smaller than those for the abnormal, which is contrary to the results of the 2nd−8th orders. The diagnostic results of the test samples are listed in Table S3 in Supplementary material 2. By comparing these results with their pathological decisions, the performance metrics of each method can be calculated and listed in Table S4 in Supplementary material 2. The metrics of the methods are finally summarized in the lower part for comparison. As the best performing one, the ENT method has a sensitivity of 66.7%, a specificity of 51.5%, an accuracy of 59.1%, a PPV of 57.9% and an NPV of 60.7%. The performance when using the 3rd−8th orders for diagnosis is unsatisfactory.

The RF distributions of the parameters by using the 3rd−8th orders have no clear demarcation between the two groups. It can be inferred from them that the HOAs are not beneficial for diagnosis, and good results may be obtained by using only low-order aberrations (LOAs). Hence, the situation by using only the 2nd order for diagnosis was studied. The RF distributions of the parameters are shown in Fig. S2 in Supplementary material 1, which are close to their corresponding distributions of the 2nd−8th orders. The diagnostic results of the test samples are listed in Table S5, and their performance metrics are listed in Table S6 in Supplementary material 2. The Comb method has the best performance, with a sensitivity of 81.8%, a specificity of 87.9%, an accuracy of 84.8%, a PPV of 87.1% and an NPV of 82.9%. Its performance is much better than that of using the 3rd−8th orders, but slightly worse than that of using the 2nd−8th orders for diagnosis.

To further investigate the performance and trend of using different orders for diagnosis, the situations by using the 4th−8th and 2nd−3rd orders for diagnosis were studied, respectively. For the former, the RF distributions of the parameters are shown in Fig. S3 in Supplementary material 1, and the diagnosed results and the performance metrics are listed in Tables S7 and S8 in Supplementary material 2, respectively. The Comb method has the best performance, with a sensitivity of 48.5%, a specificity of 75.8%, an accuracy of 62.1%, a PPV of 66.7% and an NPV of 61.0%. Its performance is also unsatisfactory. For the latter, the RF distributions of the parameters are shown in Fig. S4 in Supplementary material 1, and the diagnosed results and the performance metrics are listed in Tables S9 and S10 in Supplementary material 2, respectively. The RMS method has the best performance, with a sensitivity of 81.8%, a specificity of 90.9%, an accuracy of 86.4%, a PPV of 90.0% and an NPV of 83.3%. These metrics are completely consistent with those of using the 2nd−8th orders.

From the results of using the 3rd−8th and 4th−8th orders for diagnosis, respectively, it can be concluded that the performance of using the back orders for diagnosis is unsatisfactory. It is not necessary to further investigate the situations by using the other back orders, such as the 5th−8th orders for diagnosis. Here, the front and back orders were used to distinguish from the fixed terminologies of the LOAs and HOAs. Compared with that of using only the 2nd order, using the 2nd−3rd orders for diagnosis has improved performance, and this trend may continue. Hence, the situation by using the 2nd−4th orders for diagnosis was also studied. The RF distributions of the parameters are shown in Fig. S5 in Supplementary material 1, and the diagnosed results and the performance metrics are listed in Tables S11 and S12 in Supplementary material 2, respectively. The performance of the RMS and Comb methods is exactly the same, and both are the best. However, the former is simpler, requires fewer computations, and thus is considered as the best method. Its all the metrics listed in Table 3 are decreased relative to those when using the 2nd−3rd orders for diagnosis, indicating that further increasing the number of the orders from front to back cannot yield better results. Hence, it is not necessary to further study the situations by using other front orders, such as the 2nd−5th orders for diagnosis.

Table 3.

Final performance comparison of the respective best methods (units: %)

Method	Sen	Spe	Acc	PPV	NPV	Evaluation
RMS @ O2−8	81.8	90.9	86.4	90.0	83.3	5B0SB
ENT @ O3−8	66.7	51.5	59.1	57.9	60.7	0B0SB
Comb @ O2	81.8	87.9	84.8	87.1	82.9	1B4SB
Comb @ O4−8	48.5	75.8	62.1	66.7	61.0	0B0SB
RMS @ O2−3	81.8	90.9	86.4	90.0	83.3	5B0SB
RMS @ O2−4	78.8	87.9	83.3	86.7	80.6	0B2SB
O2,3,7,8	51.5	97.0	74.2	100.0	74.4
33 Coef	66.7	93.9	80.3	95.7	73.8

O2−8, O3−8, O2, O4−8, O2−3 and O2−4: the 2nd−8th, 3rd−8th, 2nd, 4th−8th, 2nd−3rd and 2nd−4th order aberrations, respectively; O2,3,7,8: the 2nd, 3rd, 7th and 8th order aberrations; 33 Coef: 33 Zernike coefficients;

The Bold and italics indicate the best and second-best, respectively

Performance comparison of respective best methods

Six situations by using different orders for diagnosis were studied, and the metrics of the respective best methods are summarized in Table 3 for final comparison. The RMS method by using the 2nd−8th and 2nd−3rd orders for diagnosis, respectively, has the same metrics, and all the metrics are the highest among all the methods. The amount of data is reduced when fewer orders are used, but there is no qualitative difference in the computational cost between using fewer and more orders for current computing power. However, using more orders for diagnosis can provide more comprehensive information, and avoid the randomness and instability that may occur when only a few orders are used. Therefore, it is recommended to use the 2nd−8th orders for diagnosis, and finally the RMS method by using the 2nd−8th orders is the best. On the other hand, using the 2nd−3rd orders can also achieve the same performance, and be applicable for the occasion without the ability to accurately measure the back orders. In addition, the Comb method by using only the 2nd order is the second-best, and all the metrics also exceed 80%. Using only the LOA that can be measured anywhere (e.g., in optical shops rather than eye hospitals) also has good performance, making it have a potential application even in optical shops in the future.

The metrics of the best two methods by directly using the coefficients for diagnosis in our previous work are also listed in Table 3 for comparison. The data used are exactly the same as those used in this work. Compared with them, the best method in this work has improved sensitivity, accuracy and NPV, as well as decreased specificity and PPV (but still at a high level of over 90.0%). In particular, the most important metric−sensitivity is significantly improved by 15.1%. Hence, the overall performance of using the wavefront parameters for diagnosis is much better than that of using the coefficients.

Discussion

Statistical difference of wavefront parameters

Whether there is a significant statistical difference between the two groups, is the basis for whether the parameters can be used for diagnosis. Here, inferential statistics on the two groups of each parameter were performed. The common descriptive statistics of each group are also listed in Table 4. There are significant differences in the statistics between the two groups for each parameter, except that the minimum and maximum values are close. From Fig. 3 and Table 4, respectively, it can be seen that all the frequency distributions are not close to normal, and the variances of the two groups are not close. Hence, the data do not meet the conditions of t-test, and Mann-Whitney test (MWT) needs to be adopted. The MWT is a common nonparametric test for comparing two samples with non-normal distributions. The U statistic is calculated from the rank of the two groups, and is the number of times a score in the abnormal group is larger than a score in the normal. The approximate normal test statistic Z provides an excellent approximation as the sample size grows, and the asymptotic p-value is calculated from the Z. The larger the absolute value of the U statistic and the farther the Z statistic from zero, the greater the difference between the two groups. There are significant differences in both the mean rank and sum rank between the two groups for each parameter. The obtained U statistic is large, the Z statistic is far from zero, and the exact and asymptotic p-values are zero for each parameter. Hence, the MWT result of each parameter is that there are significant statistical differences between the two groups, providing a solid foundation for this work. On the other hand, in general, the MWT is used for small samples, while t-test should be preferred for large samples even if their data do not meet the above conditions. As an additional verification, the t-test on the two groups of each parameter was also performed, and the obtained result was consistent with the above.

Table 4.

Statistical results of two groups for three parameters by using descriptive statistic and MWT methods, respectively

Descriptive statistics
Parameter	Group	Mean	SD	Variance	Minimum	Q1	Median	Q3	Maximum
RMS	Normal	5.96	2.85	8.11	0.46	4.42	5.99	7.60	20.52
	Abnormal	3.70	3.93	15.42	0.48	1.18	2.07	4.74	20.94
PV	Normal	23.70	10.67	113.78	2.57	17.93	23.64	30.00	74.27
	Abnormal	16.22	14.19	201.22	2.57	6.40	10.91	20.90	75.43
ENT	Normal	5.93	1.00	1.00	2.56	5.77	6.18	6.53	7.92
	Abnormal	4.85	1.30	1.69	2.65	3.83	4.60	5.85	7.94

Mann-Whitney test
Parameter	Group	Mean rank	Sum rank	U	Z	Exact prob>|U|	Asymp prob>|U|	Result (@ significance level of 0.05)
RMS	Normal	379.48	113464	68614	10.17	0	0	Being significantly different
	Abnormal	234.38	72891
PV	Normal	373.00	111527	66677	9.28	0	0	Being significantly different
	Abnormal	240.60	74828
ENT	Normal	379.74	113542	68692	10.20	0	0	Being significantly different
	Abnormal	234.13	72813

SD: standard deviation; Q1: 1st quartile; Q3: 3rd quartile; U: U statistic; Z: approximate normal test statistic; Exact prob: exact p-value; Asymp prob: asymptotic p-value

Impact of aberration orders on results

The frequency distributions of the parameters by using the 3rd−8th and 4th−8th orders, as shown in Figs. S1 and S3 in Supplementary material 1, respectively, indicate that the value corresponding to the high RF region in the abnormal group is slightly larger than that in the normal for each parameter. Hence, the statistical value of the back order in the abnormal group is greater than that in the normal, which is consistent with the existing result [22–24]. The distributions by using the 2nd, 2nd−3rd and 2nd−4th orders, as shown in Figs. S2, S4 and S5 in Supplementary material 1, respectively, reveal that the value corresponding to the high RF region in the normal group is significantly greater than that in the abnormal for each parameter. Hence, the statistical value of the front order in the normal group is greater than that in the abnormal. Moreover, for the distributions by using the 2nd−8th orders shown in Fig. 3, the value corresponding to the high RF region in the normal group is also significantly larger than that in the abnormal for each parameter, indicating the small contributions of the back orders to the distributions. Therefore, the frequency distributions are mainly determined by the front orders.

For the front orders, such as the 2nd, 2nd−3rd and 2nd−4th orders, the distributions of each parameter are significantly different between the two groups, and their contributions to the diagnostic performance are also significant. For the back orders, such as the 3rd−8th and 4th−8th orders, the distributions of each parameter between the two groups are relatively close, and their effects on the differentiation cancel each other out; thus, their performance is unsatisfactory. Moreover, the performance of using the 2nd−8th and 2nd−3rd orders for diagnosis, respectively, is completely consistent, indicating that the contributions of the back orders to the performance are lower.

Performance comparison of wavefront parameters

The situation numbers of the RMS, PV, ENT and Comb methods as the respective best are 3, 0, 1 and 3, respectively. For the situation by using the 2nd−4th orders, the metrics of the RMS and Comb methods are exactly the same, but the former is simpler and is regarded as the best. Hence, the methods from the best to worst performance are the RMS, Comb, ENT and PV methods sequentially. Further focusing on the situation by using the 2nd−8th orders established as the best, as shown in Table 2, the numbers of the best and second-best metrics are 5 and 0 for the RMS, 0 and 0 for the PV, 1 and 0 for the ENT, and 1 and 4 for the Comb methods, respectively. The methods from the best to worst performance are also the same as the above. The distribution trends of the PV and RMS are relatively similar in each situation, but the PV considers only the extremes of the wavefront, which is too simple to be beneficial for the diagnosis. The Comb method combines the results of the other methods, but has an even worse performance due to the low performance of the PV method. The ENT method has been successfully applied in various fields, but has unsatisfactory performance in this work. The reason for this may be as: the ENT distribution of the abnormal group is not concentrated, and there are also many abnormal samples in the region dominated by the normal group. This also may be due to the intense fluctuations in the ENT distributions, but smoothing them still cannot improve the performance.

Limitations and strengths

As the best performing one, the method by using the RMS parameter and the 2nd−8th orders for diagnosis has high specificity and PPV, as well as moderate accuracy and NPV. Its sensitivity is not high enough, and there is still a gap from achieving high-sensitivity detection of abnormal eyes. However, it is a reliable result obtained from human data collected in clinical practice rather than from animal models, theoretical models or simulations. It also cannot be ruled out the possibility that the method itself does not have higher sensitivity, accuracy and NPV, and only be suitable as a preliminary diagnostic method.

The main limitations are as follows. (1) The sample size is not large enough. In particular, the sample size of the test groups is small, which may result in inaccurate performance metrics. The lack of using external data to validate the clinical generalizability is also a drawback; (2) The generalization of the results still needs to be further validated. There will have biases in the aberrations measured by the aberrometers with different principles, or with the same principle but different models. To avoid the biases and other unknown influences, the data used in this work were collected with the same model of aberrometer. The results of the data collected by other aberrometers still need to be investigated; and (3) The diagnostic and performance evaluation methods are simplistic. A major drawback is that the diagnostic result is qualitative (only providing the result of normal, abnormal, or undetermined), rather than quantitative (without providing the probability of a subject being diagnosed as a certain category). The validity of the results still needs to be verified, due to the failure to use the methods commonly used in the medical field, such as receiver operating characteristic (ROC) curve analysis.

As a safe, simple and low-cost method for preliminary diagnosis, it still has research value thanks to its remarkable advantages. Its principle is simple, and any complex physical or medical theory is not involved. The data processing is simple, and any complex mathematical calculation or computer programming is not involved. Only the ocular aberrations are utilized, and any additional device, operation or cost is not needed. Furthermore, the RMS calculation can be greatly simplified by directly using the coefficients based on the properties of the Zernike polynomials, as presented in Supplementary material 3. Hence, it can be easily adopted by any team, even in the regions and countries with limited medical conditions and low medical levels.

In addition to specialized eye hospitals, it will also be potentially applied to optometry for community screening. The possible implementation ways include: integrating it into aberrometers to provide aberration and preliminary diagnostic results simultaneously; developing an application integrated into mobile terminals to read aberrations and then provide a preliminary diagnostic result. How to handle undetermined cases in practice is an issue that must be considered. Although there were no undetermined case in the diagnostic results of all the test samples by both the best and alternative methods, it is inevitable that it will occur in a large number of applications. Considering the serious consequences of missed diagnosis of abnormal eye, to be prudent, it is recommended to classify all undetermined cases as abnormal.

Future works

Future research will mainly focus on the following three aspects. 1) Obtaining a larger number of samples with more accurate classification. (1) More aberration data should be used for statistical analysis to make the RF distributions closer to actual situations; (2) The pathological decisions of all the samples should be as accurate as possible, i.e., the classification of the normal and abnormal groups should be accurate, for which independent diagnoses by multiple experts are needed. The samples with uncertain or controversial decisions need to be excluded; and (3) More test samples should be used to make the performance more accurate. The first two are beneficial for improving the correctness of the diagnostic model, which is the basis; and the last one is beneficial for accurately evaluating the performance; 2) Investigating the results of the aberration data collected by other aberrometers, i.e., the generalization of the method. Due to the involvements of different aberrometers and a large number of patients available for data collection, this work is usually performed only in large eye hospitals. It is suggested that the teams with the conditions use the ideas and methods proposed in this work to obtain the statistical distributions of their own data, and then perform the diagnostic research based on them; and 3) Seeking more reasonable diagnostic and performance evaluation methods, especially those widely recognized in the medical field. For instance, if the diagnostic method can be improved from qualitative to quantitative, the probability of the diagnostic result can be obtained, and then the ROC curve and its related parameters can be used to quantitatively evaluate the performance. Combining with the currently successful artificial intelligence technology, e.g., applying machine learning on the frequency distributions, is also a highly promising research direction.

Conclusion

Using wavefront parameters − the RMS, PV and ENT for the preliminary diagnosis of eye health has been demonstrated to be possible. For the normal and abnormal groups of each parameter by using the 2nd−8th orders, the data distributions are significantly different via both the MWT and t-test, and the frequency distributions are also obviously different. The recommended method by using the RMS and the 2nd−8th orders for diagnosis has the best performance, with a sensitivity of 81.8%, a specificity of 90.9%, an accuracy of 86.4%, a PPV of 90.0% and an NPV of 83.3%. The alternative method by using the RMS and the conveniently measured 2nd−3rd orders for diagnosis also achieves the same performance as the above. The latter will be potentially applicable in the occasion without the ability to measure the back orders. The methods from the best to worst performance are the RMS, Comb, ENT and PV methods sequentially.

Abbreviations

RMS

Root-mean-square

HOA

High-order aberration

PV

Peak-to-valley

ENT

Entropy

NTS

Normal test sample

ATS

Abnormal test sample

RF

Relative frequency

TP

True positive

FP

False positive

TN

True negative

FN

False negative

Spe

Specificity

Sen

Sensitivity

Acc

Accuracy

PPV

Positive predictive value

NPV

Negative predictive value

DY

Diagnostic yield

Comb

Combined diagnosis by using the diagnostic results of RMS, PV and ENT parameters

LOA

Low-order aberration

MWT

Mann-Whitney test

Author contributions

YY: conceptualization, formal analysis, funding acquisition, investigation, methodology, project administration, resources, supervision, validation, visualization, writing - original draft, and writing - review & editing. SC: software, investigation and validation. XY: formal analysis, software and visualization. HC: funding acquisition, resources and writing - review & editing. FY and QH: data curation and investigation. All authors reviewed the manuscript.

Funding

This work was supported by the National Key Research and Development Program of China (2022YFC2404302) and the National Natural Science Foundation of China (62105337).

Data availability

The data used to support the findings of this study are available from the corresponding author upon request, but the authors do not have permission to share ocular aberration data.

Declarations

Ethics approval and consent to participate

This study was conducted in accordance with the Declaration of Helsinki, and the international standards of data privacy. This is a register-based study, and the approval of an ethical committee is not needed according to Chinese legislation. All the ocular aberration data were anonymized before being transferred to us for processing, which precluded any back-tracing of the identity, and the informed consent from the patients was not necessary.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.