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The current popularity of excimer laser refractive surgery suggests a need for continued research and refinements to further improve clinical outcomes. A fundamental limitation of current clinical systems is the lack of real-time feedback specifically addressing the laser-tissue interactions as directly related to laser ablation rates. This paper reports data to assess the feasibility of a novel approach that holds promise as a real-time feedback scheme based on comparison of the incident and reflected laser pulse waveforms, as quantified using a cross-correlation algorithm. The approach is evaluated for ablation of bovine cornea over a range of clinically relevant laser fluences. A linear relationship was observed between several cross-correlation metrics and the directly measured corneal ablation rate, yielding an average RMS predictive error of 3.9% using a 25-shot average reflected waveform. Assessment of the cross-correlation approach for single-shot ablation data revealed a brief transient corresponding to the first few laser pulses, which is attributed to a slight hydration gradient near the surface of the de-epithelialized cornea. Clinical refractive data are necessary to assess the precision of this approach for actual refractive surgery.
©2011 Optical Society of America
1. Introduction
Excimer laser refractive surgery continues to play a leading role in laser-based refractive procedures, where numerous clinical systems routinely provide the desired clinical outcome with minimal side effects [1,2]. However, recent research has continued to address the clinical implementation and outcome of excimer laser surgery [3,4], with goals of better understanding the efficacy of clinical algorithms and of ultimately improving the accuracy and precision. For example, Jiménez et al. has studied the angular dependence of laser-ablation rates in the context of a mathematical model to provide better correction of eye aberrations [5]. Previous work by Jiménez et al. explored correction factors for ablation algorithms for refractive systems with Gaussian laser beam profiles [6]. Other efforts have examined the role of ablation patterns for state-of-the-art refractive surgery systems, aiming to optimize the delivery of laser pulses to reduce spherical aberration [7]. In addition to focusing on the corrective algorithms and higher-order aberration, research has addressed the laser frequency of the ablation process to ensure that thermal damage or ablation plume interactions do not adversely affect the desired ablation rate or ablation outcome [8,9], thereby allowing for reduced treatment times. Clearly the means to address precision and accuracy of clinical laser systems remains important.
While advanced treatment algorithms and higher-order models continue to improve clinical refractive surgery systems, an alternative research track is needed to provide the necessary real-time feedback for directly addressing actual patient-to-patient variations in the tissue ablation rate. Currently, even state-of-the-art excimer laser systems still rely on average (i.e., non-patient specific) ablation rates. Variations in actual patient ablation rates may stem from the complex interactions of the excimer laser pulse with corneal tissue, including dynamic photochemical processes that are coupled to the local nature of the corneal matrix [10]. For example, Fisher and Hahn developed a numerical model of 193-nm corneal tissue ablation that incorporates a dynamically changing absorption coefficient and enables exploration of corneal hydration effects [11]. Previous studies have also addressed deviations of actual corneal tissue ablation rates from those predicted by a simple Beer-Lambert model [12–15]. Such laser-tissue interactions during corneal ablation are often manifested in the tissue optical properties, including dynamic perturbations, which can subsequently affect the shape of the reflected laser beam profile. The current paper seeks to explore the incident and reflected waveforms as a means to provide real-time feedback during corneal refractive procedures.
The typical ArF excimer laser pulse has a full width (i.e., leading edge to trailing edge) of about 10 to 15 ns. Under tissue ablative conditions, however, the reflected pulse shape (waveform) typically shows a marked attenuation in intensity over the latter half of the pulse (i.e., on the trailing edge of the peak). This effect is commonly referred to as truncation, and arises due to changes in the optical properties (e.g., reflection and absorption) of the tissue during interaction with the deep-UV excimer laser pulse. Specifically, previous studies have ruled out ablation plume effects (i.e., particle scattering and/or absorption) and shown that changes occur within the corneal tissue [16]. Such changes are attributed directly to photochemical effects in the tissue matrix, which alter the tissue’s optical properties [11]. Researchers have attempted to quantify the truncation effect using the ratio of the pulse width (integrated peak area) of the reflected waveform to that of the incident waveform [17–19]. While successfully documenting the truncation phenomenon, the use of the integrated peak ratio as a metric to quantify laser-tissue interactions and related ablation rates suffers from a lack of sensitivity, as moderate to subtle changes in truncation are largely lost in the integration of total waveform energy. The current research was primarily motivated by the goal of using the observed truncation of the reflected waveform as a means to quantify the corneal tissue ablation rate. Specifically, the method of cross-correlation was developed to emphasize the truncation effect in a suitable, quantifiable metric of real-time tissue ablation. To this end, preliminary experiments were performed using whole bovine eyes.
2. Experimental methods and procedures
2.1 Experimental setup
The configuration for these experiments is shown in Fig. 1 . Laser-beam temporal profiles (waveforms) were collected as whole bovine eyes (see details below) were ablated using an excimer laser. The laser used in these experiments was a research version of the clinical LADARVision laser system made by Alcon. It is a 193-nm ArF excimer laser, with 10-ns pulse width, 5-mJ maximum pulse energy, and 100-Hz maximum repetition rate. In these experiments the laser was operated at approximately 2 Hz and the full-width spot size at the focal point was 0.97 mm2. A UV-grade quartz wedge was aligned at 45° with respect to the beam path to reflect approximately 5% of the energy, which was directed through a 193-nm interference filter to a fast-response (200-ps rise time), UV-sensitive photodetector (Model R1193U-02, Hamamatsu Corp., Bridgewater, NJ). This reference beam provided what is referred to as the incident waveform, and represents the original laser pulse shape.
Fig. 1 Experimental configuration: EL: excimer laser; QW: quartz wedge; EM: excimer mirror; ND: neutral density filter; IF: interference filter; PD: photodetector; PM: pierced mirror; BE: bovine eye; FL: focusing lens; DO: digital oscilloscope.
Approximately 94% of the laser energy was transmitted through the quartz wedge, reflected by an excimer (193-nm) dichroic mirror, and directed through a pierced mirror to a bovine eye. Reflected light from the surface of the eye during the ablation event was reflected by the pierced mirror and then focused by a lens, through a 193-nm interference filter, onto a second, matched UV-sensitive photodetector. The signals (i.e., temporal waveforms) from both photodetectors were recorded and saved using a 2.5 GS/s, 500-MHz digital oscilloscope (Model 9361, LeCroy Corp., Chestnut Ridge, NY). Neutral density filters were used as necessary to attenuate both the incident and reflected beams to ensure signal linearity and prevent detector saturation, thereby preserving the true, temporal waveform shape.
2.2 Experimental procedure
The overall experimental procedure involved both the acquisition of laser-beam waveforms (incident and reflected) during ablation, and the determination of the average tissue ablation rate for each ablation site. Acquisition of the laser waveforms during ablation is detailed above in Section 2.1. The true ablation rate was determined using a method that was described in detail in a previous paper [20]. Briefly, warm liquid wax was applied with a dropper to an ablation crater immediately after laser ablation. Once dry, the wax impression of the crater was carefully removed and then measured using a white-light interferometer to determine the depth of the ablation crater. The average per-shot ablation rate was then calculated by simply dividing the total crater depth by the number of laser pulses used to create the ablation crater. For this study, the laser beam profile was a truncated Gaussian that approached a top-hat shape, and all reported ablation rates correspond to the relatively flat central depth of the ablation crater. The method was previously shown to provide a high-fidelity profile of the ablation crater and to correlate very well with direct measurement of ablation craters for excimer laser ablation of polymers [20]. Furthermore, the method provided direct measurements of bovine cornea ablation-rate data in excellent agreement with clinical ablation rates for identical laser conditions.
A total of eight bovine eyes were used in this study. Whole bovine eyes were obtained from a slaughterhouse, immediately after sacrifice, and stored at ambient temperature (indoor) of about 23°C in small-volume, saline-containing plastic bags until used. The cornea surfaces were not immersed in saline, but rather, the saline was added to quickly establish saturation in the sealed bags, thereby eliminating any drying of the cornea. Experiments were performed on the day of animal sacrifice, typically within 4 to 6 hours of removal. The procedure for each eye was identical, and consisted of the following steps:
- 1. The epithelial layer was carefully removed from the eye by manually scraping across the corneal surface with a scalpel edge.
- 2. The bovine eye was placed in a fixture and aligned such that the laser beam was normal to and focused at the corneal surface.
- 3. A total of 25 ablating laser pulses were delivered to the bovine eye, during which time incident and reflected laser waveforms were collected.
- 4. A wax impression of the ablation crater was created and set aside for later analysis.
- 5. The bovine eye was rotated to expose a fresh, unaffected area for the next ablation. Laser waveforms were collected as another series of 25 ablating laser pulses were delivered, and a wax impression was created of the new crater.
- 6. Step 5 was repeated for a third ablation site (i.e., three sites per cornea). Spacing between the three ablation sites exceeded the size of a given wax application.
The laser fluence used for ablation was varied from eye to eye, and ranged from about 290 to 465 mJ/cm2 (2.8 to 4.5 mJ/pulse) as consistent with actual clinical values. The resulting corneal ablation rates were therefore expected to vary due to the differences in laser fluence. In addition, slight variations in corneal hydration were considered a potential source of variation in ablation rate [11,21,22]. Although not reported here in detail, corneal hydration was assessed for each bovine eye immediately before ablation at the first site and immediately after ablation at the third and final site. For each ablation site, hydration was interpolated between the bracketing values according to the time schedule of the ablation experiments. Specifically, hydration was measured using confocal Raman spectroscopy, with the setup and technique described in detail in a previous paper [23]. The ratio of a Raman band at ~3400 cm−1 corresponding to an OH stretching vibration (present in water) and another Raman band at ~2940 cm−1 corresponding to a CH stretching vibration (present in collagen) served as the hydration metric. This ratio was shown to increase for increasing corneal hydration, since the OH band grows stronger due to the increase in water content while the CH band remains relatively unchanged [23]. No overall quantitative hydration data (i.e., no absolute calibration) are reported here; however, the average Raman corneal hydration signal measured for the eyes studied was 8.6±0.4 in arbitrary units of Raman scattering intensity.
2.3 Cross-correlation function
For the time-varying laser pulse waveform represented by I(t), it is possible to determine the value of the function at any time (t), or at any point (t + τ) that is shifted (i.e., delayed) from the original time. The two values, I(t) and I(t + τ), are highly correlated as τ approaches zero, and are identical when the delay is equal to zero (τ = 0). As the delay is increased, the two values become less correlated until the point where the delay exceeds the time domain of the function I(t), and there is zero correlation between the two values. For discrete time intervals, the autocorrelation function of I(t) can be expressed by the relation in Eq. (1):
where n is the discrete time index step corresponding to a given τ.A cross-correlation function, illustrated below in Fig. 2 , is similar to an autocorrelation function, but shows the degree of correlation between two different waveforms rather than the same waveform. The normalized cross-correlation function, evaluated for discrete intervals, used in the current work is described by:
In Eq. (2), the left-hand side is a representation of the cross-correlation function between the incident (I) and reflected (R) waveforms. The first term on the right-hand side is the value of the autocorrelation function of the incident waveform for τ = 0, which is used to normalize the cross-correlation function. This normalization factor is calculated by squaring each data point of the incident waveform and summing these values, and is used to eliminate the effect of absolute signal, thereby mitigating any effects of signal fluctuations from pulse-to-pulse, site-to-site, or day-to-day. To eliminate the issue of amplitude and make the cross-correlation a function of the waveform shape only, the reflected signal was always normalized to the incident signal using the maximum amplitude. Therefore, the peak values of the two waveforms were always the same. To compensate for the longer pathlength compared to the measured incident waveforms, reflected waveforms were shifted forward in time the appropriate amount, as determined from sub-ablative laser pulse measurements on a glass slide. Once determined, this constant shift was introduced to all collected reflected waveforms prior to processing. The maximum resolution of the oscilloscope was 0.4 ns, so this was the time step used for evaluation of the discrete cross-correlation function.Fig. 2 Example incident and reflected waveforms (a) and the resulting cross-correlation function (b). Data correspond to actual waveforms recorded from bovine cornea.
Construction of the cross-correlation function is illustrated in Fig. 2. Figure 2(a) shows the incident waveform and the reflected waveform at various values of the time delay τ, and Fig. 2(b) shows the resulting cross-correlation function. For a given time delay τ, the intensity values of the incident and time-shifted reflected waveforms at each time step were multiplied together, and these values were then summed. The value of τ was then incremented, and the process repeated until the two waveforms no longer overlapped (i.e., zero cross-correlation). The cross-correlation function value for each given τ was then normalized by dividing by the autocorrelation of the incident waveform evaluated for zero time delay.
There are several quantities of interest that can be extracted from the cross-correlation function and related to the ablation rate. In the present work, two are selected as the most suitable metrics. The first metric is the initial slope, at τ = 0, and the second is the decay slope of the cross-correlation function. Figure 3 illustrates these two slopes for the cross-correlation function shown in Fig. 2(b). The initial slope was calculated using a forward finite-divided difference formula with second-order accuracy, which uses the first four data points (including at τ = 0) to quantify the derivative at τ = 0. The decay slope was calculated from a linear least-squares regression line fit to the data points spanning from τ = 3.2 ns to τ = 8.4 ns.
Fig. 3 Illustration of the initial slope (a) and the decay slope (b) of the cross-correlation function.
3. Results and discussion
As mentioned in Section 2.2, ablation experiments were carried out for a range of laser fluence/energy. Figure 4 shows measured ablation rates as a function of laser pulse energy. Although the data may also have been affected slightly by variations in corneal hydration, hence the scatter, Fig. 4 shows that ablation rate increases linearly with increasing laser pulse energy over this range. There are two data points that are identified as outliers (greater than 0.1 μm/pulse deviation from the linear trend line), and they are circled in Fig. 4. Although included in Fig. 4 for completeness, these two points were omitted for all subsequent analysis.
Fig. 4 Ablation rate vs. laser pulse energy for bovine eye ablation experiments. The two circled data points were considered outliers.
As defined in Section 2.3, both the initial slope and the decay slope of the cross-correlation between incident and reflected waveforms were considered the metrics of interest for this study. In addition, the ratio of the initial slope to the decay slope was also investigated. Figure 5 shows the relationships between these three quantities and the measured ablation rates for the bovine eyes used in this study.
Fig. 5 Initial slope (a), decay slope (b), and ratio of initial slope to decay slope (c) of cross-correlation function vs. bovine eye ablation rate.
For each of the metrics, there appears to be a strong linear correlation to the ablation rate. In terms of an ablation-rate predictor, it appears that the decay slope and the slope ratio (initial:decay) are both good candidates for real-time measurement of ablation rate. The sensitivity of the slope ratio is greater than that of the decay slope (i.e., steeper regression slope), while the degree of correlation to the ablation rate is greater for the decay slope (i.e., greater regression r2 value). The initial slope does not appear as good of a metric, since both the sensitivity (i.e., regression slope) and the degree of correlation (i.e., r2 value) are not as large as for the other two quantities. Overall, the ablation rate seems to be very well predicted by both the decay slope and the slope ratio as implemented using the cross-correlation for the 25-shot averages.
The results shown in Fig. 5 could have a significant impact on corneal refractive surgery with regard to providing a suitable metric for real-time feedback. The ablation rate varied over a relatively large range (0.8–1.2 µm/pulse) in these experiments due to variations in laser pulse energy, and possibly due to corneal hydration, while the cross-correlation function provided outputs that can clearly be directly correlated to ablation rate over the entire range. It is important to note that the current metrics (cross-correlation slopes) and the actual ablation rates were measured experimentally and independently. No relationships between ablation rate and laser pulse energy or corneal hydration were assumed. This suggests that corneal refractive surgical systems could use cross-correlation slopes (acquired passively during surgery in real-time) to modify ablation algorithms at any time for any number of reasons, including patient-to-patient variation, variation in delivered pulse energy, and transient or site-to-site changes in corneal hydration, although this remains to be validated.
It is difficult to assess the statistical error associated with the linear correlation functions presented in Fig. 5 because no means are available to decouple the errors associated with the cross-correlation metric from those associated with the measured ablation rate. Regardless, using the linear correlation curve for the decay-slope data reveals a RMS error of 3.9% when using the individual decay-slope values to predict the corresponding realized ablation rates. An average ablation rate of 10.9 μm per spherical equivalent diopter of correction was recently reported for three clinical LASIK systems [24]. A reasonable measure of refractive success may be defined as within 0.5 diopters of the intended correction [25,26], which corresponds to about 5.5 μm of corneal tissue based on the above clinical average. Applying the 3.9% RMS error from the current experiments to the above-cited clinical ablation rate, errors of 1.3, 2.6, and 3.8 μm of corneal tissue are estimated for refractive corrections of 3, 6, and 9 diopters, respectively. Although additional experimental data are required for rigorous statistical analysis, it is expected that greater averaging of the reflected waveform data will improve the precision of the cross-correlation approach. Nonetheless, all three estimated uncertainties are within the 5.5-μm value corresponding to the 0.5-diopter criterion, suggesting the necessary precision and therefore the potential to make corrections in refractive treatments to address errors greater than 0.5 diopters.
The results presented thus far could prove useful for large variations in ablation rate on an average-shot basis. For example, a refractive surgical procedure could proceed for some number of laser pulses, at which time the refractive algorithm could be adjusted based on the average cross-correlation function over those pulses, and the procedure continued with the adjusted algorithm. This ablation-rate adjustment could be done once during the procedure, or possibly several times if necessary. Alternatively, it might be possible to adjust the algorithm continuously based on real-time feedback for each and every ablating laser pulse. Whether or not continuous, real-time adjustment is necessary or even feasible with this approach in clinical algorithms remains to be determined. How much the cross-correlation metric varies on a pulse-to-pulse basis, however, was something that could be studied in these experiments.
Figure 6 shows cross-correlation decay slopes for a sequence of 25 ablating laser pulses at one site on a bovine eye for a laser fluence of 450 mJ/cm2. This laser fluence is near the high end of the fluence range used in this study, and was chosen for the study of pulse-to-pulse variations primarily because of the high degree of laser energy stability (< ± 1% standard deviation) and the maximum depth of total ablation over the 25-shot sequence. The pulse stability is important, because it means that variations in ablation rate, and therefore variations in the cross-correlation decay slope, were not caused by changes in laser energy over this specific 25-shot sequence. The dashed horizontal line represents the value of the cross-correlation decay slope determined from the average incident and reflected waveforms over the entire pulse sequence (N = 25). It is noted that this entire data set corresponds to a single data point in Fig. 4 and 5.
Fig. 6 Cross-correlation decay slopes for a sequence of 25 ablating laser pulses at a single ablation site on a bovine eye.
With such small variation in laser energy, variations in ablation rate and therefore in the cross-correlation decay slope are likely caused by variations in actual corneal hydration with tissue depth. As mentioned in Section 2.2, corneal hydration of the bovine eyes was assessed immediately before ablation at the first site and after ablation at the third and final site. While results of the Raman measurements are not presented as part of this study, the data revealed a slight positive dependence of ablation rate on corneal hydration, meaning that ablation rate appeared to increase with increasing hydration. Such a trend is in agreement with recent modeling results that predict a direct relationship of increasing ablation rate with increasing corneal hydration [11]. The results shown in Fig. 6 are consistent with this relationship. After the epithelium is removed from a bovine eye, exposure to outside air dehydrates the corneal tissue and establishes a hydration gradient as shown previously [23]. The exposed surface, where the ablation sequence begins, is driest, with hydration gradually increasing with corneal depth until reaching a near-steady hydration level (i.e., bulk stroma hydration) away from the corneal surface.
Figure 6 shows that the cross-correlation decay slope started at its smallest value for the first laser pulse (at the dry, exposed corneal surface) and increased within a few shots to a relatively steady value (corresponding to bulk corneal hydration). This may be considered the steady-state ablation value, attributed to establishment of thermal equilibrium and/or equilibrium corneal hydration. The initial transient behavior notwithstanding, Fig. 6 shows that the decay slope was relatively stable on a pulse-to-pulse basis, and that its value for a given individual laser pulse was very similar to the average value determined from the cross-correlation of the average incident and average reflected waveforms for the entire 25-pulse sequence. Furthermore, the reduced cross-correlation slope for the first few laser pulses is considered to reflect actual changes in ablation rate due to localized, near-surface changes in tissue hydration. It is noted that one is unable to construct calibration curves similar to those in Fig. 5 on a single-shot basis, as no method exists for individual single-shot ablation measurements of corneal tissue. In practice, one might envision collecting and storing single-shot data such that average waveforms may be calculated for spatial regions after a sufficient number of incident laser pulses.
4. Summary and conclusions
When using the 193-nm ArF excimer laser for corneal tissue ablation, as in corneal refractive surgeries such as LASIK, the temporal profile (i.e., laser-pulse waveform) of the reflected 193-nm light has a truncated shape that is different from the incident waveform. This study has shown that the cross-correlation function of the incident and reflected waveforms can be used to quantify this truncation effect and relate it to the tissue ablation rate of bovine cornea. Three metrics derived from the cross-correlation function were investigated: (1) initial slope at τ = 0, (2) decay slope from τ = 3.2 ns to τ = 8.4 ns, and (3) ratio of initial slope to decay slope. The following conclusions were reached based on the current results:
- 1. All of the cross-correlation metrics were closely correlated to the corneal tissue ablation rate, each with an approximately linear relationship.
- 2. Both the decay slope and the slope ratio (initial:decay) showed promise as potential tools for real-time feedback during laser corneal ablation procedures. The initial slope was not as well correlated with, and was less sensitive to, the ablation rate compared with the other two cross-correlation metrics.
- 3. Laser corneal refractive procedures could potentially be adjusted for variations in actual ablation rate, whether due to variations in delivered laser pulse energy or in tissue hydration, on a pulse-by-pulse basis. Results show that the cross-correlation decay slope was steady for a sequence of 25 ablating laser pulses, after an initial transient over the first few pulses that is attributed to a slight hydration gradient.
- 4. Implementation into a clinical refractive laser system could allow for considerable complexity in the algorithms. For example, single-shot data could be stored for a typical flying-spot system, and then local averages could be calculated and the cross-correlation applied to each local spatial region once sufficient shots were amassed (e.g., 10 to 20). This would enable the precision of multi-shot averaging, would allow for masking of the first few surface-shots (see Fig. 6), and provide for spatially-resolved algorithm corrections in a manner consistent with the concept of custom-ablation profiling.
Acknowledgements
This work was supported in part by a grant from Alcon Research, Ltd. The views and opinions expressed in this article are those of the authors.
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