May 2011
Volume 52, Issue 6
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Cornea  |   May 2011
Simulation of the Impact of Refractive Surgery Ablative Laser Pulses with a Flying-Spot Laser Beam on Intrasurgery Corneal Temperature
Author Affiliations & Notes
  • Mario Shraiki
    From the Computer Sciences Group, University of Applied Sciences, Darmstadt, Germany;
    SCHWIND eye-tech-solutions, Kleinostheim, Germany; and
  • Samuel Arba-Mosquera
    SCHWIND eye-tech-solutions, Kleinostheim, Germany; and
    Grupo de Investigación de Cirugía Refractiva y Calidad de Visión, Instituto de Oftalmobiología Aplicada, University of Valladolid, Valladolid, Spain.
Investigative Ophthalmology & Visual Science May 2011, Vol.52, 3713-3722. doi:10.1167/iovs.10-6706
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      Mario Shraiki, Samuel Arba-Mosquera; Simulation of the Impact of Refractive Surgery Ablative Laser Pulses with a Flying-Spot Laser Beam on Intrasurgery Corneal Temperature. Invest. Ophthalmol. Vis. Sci. 2011;52(6):3713-3722. doi: 10.1167/iovs.10-6706.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Purpose.: To evaluate ablation algorithms and temperature changes in laser refractive surgery.

Methods.: The model (virtual laser system [VLS]) simulates different physical effects of an entire surgical process, simulating the shot-by-shot ablation process based on a modeled beam profile. The model is comprehensive and directly considers applied correction; corneal geometry, including astigmatism; laser beam characteristics; and ablative spot properties.

Results.: Pulse lists collected from actual treatments were used to simulate the temperature increase during the ablation process. Ablation efficiency reduction in the periphery resulted in a lower peripheral temperature increase. Steep corneas had lesser temperature increases than flat ones. The maximum rise in temperature depends on the spatial density of the ablation pulses. For the same number of ablative pulses, myopic corrections showed the highest temperature increase, followed by myopic astigmatism, mixed astigmatism, phototherapeutic keratectomy (PTK), hyperopic astigmatism, and hyperopic treatments.

Conclusions.: The proposed model can be used, at relatively low cost, for calibration, verification, and validation of the laser systems used for ablation processes and would directly improve the quality of the results.

Laser refractive surgery is an ophthalmic technique used to reshape the anterior corneal surface for the correction of refractive errors. In this technique, a laser beam is applied to the corneal surface for the ablation of tissue. 1 Using a flying laser beam, the surgeon has better control over laser energy delivery at each corneal position and therefore a greater demand to reproduce details accurately is put on the laser systems. 2 To implement wavefront-based customized ablation, a surgical laser system must be capable of reproducing the details of complex wavefront-driven ablations while reducing the incidence of high-order aberrations after surgery. 3,4 It is well known that a successful surgery depends on the correct design of an ablation profile, precise delivery of laser energy to the corneal position, and reliable understanding of the corneal tissue response. 
Since the introduction of laser refractive surgery, technology has evolved significantly. Today's technology uses sophisticated algorithms and optimized tools in the planning and proposes the challenge of improving surgical outcomes in visual acuity and night vision. At the same time, patients are better informed with regard to the potential of laser refractive surgery, raising the quality requirements demanded by patients. 
When refractive surgery is used, an increase in corneal temperature during surgery has been observed clinically, 5 10 as well as in the laboratory. 11,12 The rise in corneal temperature and its associated consequences are not yet fully understood. 
The available methods allow for the correction of refractive defects, such as myopia, 13 hyperopia, 14 and astigmatism. 15 Achieving accurate clinical outcomes and reducing the likelihood of a retreatment procedure are major goals of refractive surgery. For that, accurately calibrated lasers are essential. 
A large number of factors influence laser ablation and outcome. Among them, laser energy delivery technique, 16,17 ablation decentration and registration, 18,19 eye tracking, 20,21 flap, 22 physical characteristics of ablation, 23 29 wound-healing, and biomechanics of the cornea 30 33 have been explored to predict or explain the clinically observed discrepancy between intended and actual outcomes. The quantification of influence of these factors is important for providing the optimal outcome in refractive surgeries. 
The main purpose of this article is to explore details of surgery models incorporating most of the factors mentioned above, with special emphasis on the analysis of the thermal load during treatment. With a flying-spot laser, the resulting ablation profile must be deconvolved into a series of shot positions, often requiring more than 10,000 shots for the surgery. The AMARIS (SCHWIND eye-tech-solutions, Kleinostheim, Germany) proprietary shot pattern algorithm is used for this deconvolution. For a comprehensive simulation model, the contribution of the progressive nature of ablation to the final surgery outcome is included. 
Although the shot pattern and shot sequence algorithms of the SCHWIND AMARIS are used in the model, the general results produced by the model would be similar if other, generic, flying-spot algorithms were used in place of the system's proprietary algorithms, provided that the generic algorithms distributed the shots in a sequence that removes the tissue smoothly and progressively across the whole cornea over the course of the ablation. However, some of the numerical results reported here are believed to be restricted to the proprietary shot pattern or shot sequence algorithms used in this work. 
Next, we use our model to explain the causes of the changes in corneal temperature produced by refractive surgery. Our results will be important both in the optimization of laser system parameters for refractive surgeries and in preadjusting an ablation profile for better surgical outcomes. 
Materials and Methods
Calculation of the Thermal Load per Laser Pulse
Although UV radiation is considered “cold” radiation, it is only because the thermal relaxation time of the molecules used is shorter than the thermal denaturation time. There is another kind of laser interaction with the material. The photon energy is high enough to crack the C-bounding. Only part of the energy is left to heat the remaining tissue. This kind of interaction is responsible for the term cold ablation. ArF laser ablation has few thermal side effects because the optical penetration depth is shallow (≈0.35 μm), and the laser pulse duration is much shorter than the thermal diffusion in a layer of this thickness. However, this does not imply that the laser UV radiation does not increase the temperature locally. 
The thermal load of a single-pulse model describes the variation of temperature in the cornea at a given position after receiving a laser spot. The thermal load model is characterized by an immediate effect on the initial temperature in the cornea at a given position, proportional to the beam energy density at that point. 34 The temperature-cooling model used is an immediate effect model, because the thermal absorption time is very short and the local heat propagation can be regarded as infinity velocity propagation, as when using the Fourier models. 35  
Thermal load occurring at the end of a LASER pulse can be estimated by 11,36   where I is the radiant exposure; R is the reflectivity; α is the absorption coefficient; ρ is the density; and C is the specific heat. 
From the blow-off model (derived form Beer-Lambert's law 37 ) we know that the real energy density absorbed at that point determines the ablation depth:   where d is the actual depth per pulse, and I Th is the corneal ablation threshold. 
Replacing the thermal load in equation 1:   For our simulations we set the density at ρ = 1192 g/L, the specific heat capacity at c = 2.93 J/(g K), the ablation threshold at I Th = 44 mJ/cm2, and the absorption coefficient of the cornea at 193 nm α = 3.57 × 106 m−1 (Fig. 1). 
Figure 1.
 
Ablative spot and thermal load as calculated by equation 3.
Figure 1.
 
Ablative spot and thermal load as calculated by equation 3.
Interaction of the Laser Beam at Oblique Incidence
The interaction of 193-nm excimer laser radiation and corneal tissue is a complex process, involving both ultraviolet photochemistry and rapid thermal decomposition. 38 With the flying-spot laser system, the corneal ablation behavior is mainly governed by the relationship between the per-pulse tissue ablation depth and the radiant exposure (energy per illuminated area) to the incident laser radiation. 
At the laser–cornea interaction, the incident laser beam splits into a propagated/absorbed beam inside the cornea and a reflected beam. The reflectance and absorption vary with angle. Many studies have shown that the efficiency of the laser changes across the cornea, primarily because of the enlargement of the laser spot as it moves away from the corneal apex and because of differences in reflected/absorbed energy as a function of the angle of incidence. 
This interaction is accounted for in our model according to equation 3, and it has is estimated similar to the method published in the recent literature. 39,40  
Cooling of the Cornea with Time
The temperature-cooling model describes the variation of temperature in the cornea at a given position over time. The temperature-cooling model is characterized by transient smoothing of the initial temperature in the cornea at a given position asymptotic to the temperature of the environment tending toward stable equilibrium. 41 The temperature-cooling model used is a simple diffusion model that can be obtained from simple propagation models by assuming that the environment is much bigger than the tissue where the diffusion mainly occurs. 42  
Newton's law of cooling states that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings. 43 This form of heat loss principle, however, is not very precise; a more accurate formulation requires an analysis of heat flow based on the heat equation in an inhomogeneous medium. This simplification can be applied so long as it is permitted by the Biot number. 
Nevertheless, it is easy to derive from this principle the exponential decay of temperature of a body:   where T env is the asymptotic temperature (temperature of the environment), and τr is the so-called thermal relaxation time. The thermal relaxation time depends on the geometry of the heated volume and is different for objects embedded in heat-conducting material, with a surface in air. It is a material-specific quantity depending on the absorption coefficient, the thermal conductivity, the density, and the heat capacity (τr = ρc/4α2κ). 36  
Alternatively,   For the simulations, we set the thermal relaxation time of the cornea at τr = 0.14 seconds (Fig. 2). 
Figure 2.
 
Temperature change of corneal tissue for 200 flat-top laser pulses illuminating the complete corneal surface with a thermal load of 0.6 K per pulse determined for different repetition rates, as calculated by equation 5.
Figure 2.
 
Temperature change of corneal tissue for 200 flat-top laser pulses illuminating the complete corneal surface with a thermal load of 0.6 K per pulse determined for different repetition rates, as calculated by equation 5.
Heat Propagation with Time across the Corneal Surface
The heat propagation model describes the variation of temperature in a given region over time. In the special case of heat propagation in an isotropic and homogeneous medium in the three-dimensional space, 44 this equation is:   where T is temperature as a function of space and time and k is the thermal diffusivity, a material-specific quantity depending on the thermal conductivity, the density, and the heat capacity (k = κ/ρc). 
Solutions of the heat equation are characterized by a gradual smoothing of the initial temperature distribution by the flow of heat from warmer to colder areas of an object. Generally, many different states and starting conditions will tend toward the same stable equilibrium. Therefore, to reverse the solution and conclude something about earlier times or initial conditions from the present heat distribution is very inaccurate, except over the shortest of time periods. 45  
The heat equation governs heat diffusion, as well as other diffusive processes, such as particle diffusion or the propagation of action potential in nerve cells. 46  
The heat equation is, technically, in violation of special relativity, because its solutions involve instantaneous propagation of a disturbance. 47  
We can approximate the derivatives by finite differences. If we expand the function T by using the Taylor formula, we get   We consider, for any single point at the temperature matrix, the influence coming from its eight primary neighbors. We should then consider a specific case whenever we are at a corner or edge of the matrix. 
Because of the discrimination and the finite elements approach used, we must also consider whether Δt is “short enough” to ensure gradual smoothing of the initial temperature distribution by the flow of heat from warmer to colder areas of an object. 
Considering, as well, specific cases for corners and edges of the matrix, finally   After these calculations, the result will be a heat propagated at the t 1 temperature matrix. 
For our simulations we used the thermal diffusivity k = 1.43 × 10−7 m2/s (Fig. 3). 
Figure 3.
 
Spatial heat propagation across corneal tissue for a 1-mm diameter Gaussian laser pulse with a thermal load of 0.6 K per pulse determined for different propagation times, as calculated by equation 8.
Figure 3.
 
Spatial heat propagation across corneal tissue for a 1-mm diameter Gaussian laser pulse with a thermal load of 0.6 K per pulse determined for different propagation times, as calculated by equation 8.
The Virtual Laser System
For our simulations, we developed a virtual laser system (VLS) that includes many functions for simulating different physical effects of an entire surgical process (e.g., local loss of ablation efficiency due to non-normal incidence, local thermal load of a single pulse, local loss of thermal load due to non-normal incidence, global cornea cooling process by the air, global heat propagation across the cornea, local curvature progression during the treatment, global hydration progression during treatment, local pachymetric progression during treatment, and global gain of efficiency due to changes in hydration, eye movements), simulating the shot-by-shot ablation process (with or without considering the latency time of the eye-tracker to ablation) based on a modeled beam profile. 
These physical effects are not independent of one another; rather, they influence one another in a complex manner. 
The simulations at the VLS are based on local physical effects occurring immediately after a pulse reaches the cornea, followed by global physical effects occurring between two laser pulses reaching the cornea. Local physical effects (e.g., local loss of ablation efficiency due to non-normal incidence, local thermal load of a single pulse, local loss of thermal load due to non-normal incidence, local curvature progression during the treatment, and local pachymetric progression during the treatment) are confined to the areas on the cornea that are the size of a single laser spot, whereas global physical effects (e.g., global cornea cooling process by the air, global heat propagation across the cornea, global hydration progression during treatment, global gain of efficiency due to changes in hydration, and eye movements) extend over the whole corneal surface. 
Analysis of the Thermal Load of Actual Refractive Surgery Treatments
To virtually analyze the thermal load of actual refractive surgery treatments, the CAM software (SCHWIND eye-tech-solutions, Kleinostheim, Germany) was used to plan the ablations, which were ablated with the AMARIS 750S laser (SCHWIND eye-tech-solutions, Kleinostheim, Germany). 
Results
The thermal load of a single pulse depends on the applied radiant exposure, the corneal reflectivity, the thermal conductivity, the density, and the heat capacity. As the radiant exposure increases, the thermal load of a single pulse increases. Even below the ablation threshold, laser pulses contribute to the thermal load, as shown in Figure 4
Figure 4.
 
The thermal load of a single pulse mainly depends on the applied radiant exposure. As the radiant exposure increases, the thermal load of a single pulse increases. Even below the ablation threshold, laser pulses contribute to the thermal load.
Figure 4.
 
The thermal load of a single pulse mainly depends on the applied radiant exposure. As the radiant exposure increases, the thermal load of a single pulse increases. Even below the ablation threshold, laser pulses contribute to the thermal load.
As the loss of ablation efficiency decreases toward the periphery, the thermal load decreases as well. This change is shown in Figure 5 for different corneal radii of curvature. 
Figure 5.
 
Reduced thermal load due to ablation efficiency as a function of the radial distance. The thermal load was calculated for different radii of curvature and for an excimer laser with Gaussian radiant exposures of 300 mJ/cm2 and 1-mm diameter spot size. Note the reduced thermal load in peripheral locations and with steeper corneal curvatures.
Figure 5.
 
Reduced thermal load due to ablation efficiency as a function of the radial distance. The thermal load was calculated for different radii of curvature and for an excimer laser with Gaussian radiant exposures of 300 mJ/cm2 and 1-mm diameter spot size. Note the reduced thermal load in peripheral locations and with steeper corneal curvatures.
Assume that the system fires laser pulses at a 750-Hz repetition rate, but at the same time, the sequence of the pulses is resorted in a way that limits a maximum local frequency of 39 Hz. The upper limit of the thermal load of a treatment in these circumstances is depicted in Figure 6
Figure 6.
 
Upper limits for the thermal load for a 20-second treatment duration. The thermal load was calculated based on the repetition rate of the system (750 Hz) and based on the local frequency limit of the system (39 Hz) for an excimer laser with thermal load per pulse of 0.6 K. Notice that without local frequency controls for this system, temperature increases of up to 65 K are predicted, but the local frequency control limits the thermal load below 4 K.
Figure 6.
 
Upper limits for the thermal load for a 20-second treatment duration. The thermal load was calculated based on the repetition rate of the system (750 Hz) and based on the local frequency limit of the system (39 Hz) for an excimer laser with thermal load per pulse of 0.6 K. Notice that without local frequency controls for this system, temperature increases of up to 65 K are predicted, but the local frequency control limits the thermal load below 4 K.
In Figure 6, the temperature at 750 Hz means that the laser fires at the same position a 750 Hz, whereas in the other situation, the laser pulse is displaced by a scanner to get a local repetition rate of 39 Hz. 
With actual pulse lists generated by the AMARIS system for typical treatments needing approximately the same number of pulses (∼15,000 pulses, i.e., ∼20 seconds treatment duration), we compared the thermal loads of the different treatment types. The results of these simulations are depicted in Figure 7
Figure 7.
 
Simulation of different treatment types with durations of 20 seconds for the AMARIS system. Notice that the thermal load of all treatments remains always below the upper limit determined by the maximum local frequency for this system, temperature increases of up to 4 K are predicted, but the simulated thermal loads range between 3.6 K for myopia treatments and 1.8 K for hyperopia.
Figure 7.
 
Simulation of different treatment types with durations of 20 seconds for the AMARIS system. Notice that the thermal load of all treatments remains always below the upper limit determined by the maximum local frequency for this system, temperature increases of up to 4 K are predicted, but the simulated thermal loads range between 3.6 K for myopia treatments and 1.8 K for hyperopia.
With the pulse lists generated using the AMARIS algorithm for typical treatments needing approximately the same number of pulses (∼15,000) but different repetition rates, we have compared the thermal loads of the different repetition rates. The results of these simulations are depicted in Figure 8
Figure 8.
 
Simulation of the effect of the repetition rate on thermal load for different treatment types with a similar number of pulses for the AMARIS system. Note that the thermal load of all treatments at all repetition rates remains below the upper limit determined by the maximum local frequency for this system (4 K).
Figure 8.
 
Simulation of the effect of the repetition rate on thermal load for different treatment types with a similar number of pulses for the AMARIS system. Note that the thermal load of all treatments at all repetition rates remains below the upper limit determined by the maximum local frequency for this system (4 K).
Discussion
Achieving accurate clinical outcomes and reducing the likelihood of a retreatment procedure depend on accurately calibrated lasers. Parallel to the clinical developments, increasingly capable, reliable, and safer laser systems with better resolution and accuracy are needed. 
If the corneal tissue is heated up beyond a certain limit, the collagen proteins denaturize, altering their normal function. Denaturizing of the corneal collagen proteins seems to occur at temperatures higher than approximately 40°C, introducing the risk of thermal damage. 8 Considering that normal ocular surface temperature lies at approximately 33°C, the thermal load of the treatments should be limited to approximately 7°C. 
The repetition rate with which a laser pulse with a given radiant exposure can be continuously delivered onto the same corneal tissue location without denaturizing the proteins is defined as the maximum allowed local frequency. With this given radiant exposure, firing at higher local frequencies will result in the denaturizing of proteins and finally in suboptimal ablation results. 
An excessive thermal load on the cornea is an effect that should be avoided in commercial laser systems that use sophisticated algorithms that cover most of the possible variables. This study provides analytical expressions for calculation of the thermal load on the cornea. The model directly considers applied correction and corneal geometry, including astigmatism and corneal toricity, as well as laser beam characteristics and ablative spot properties. A separate analysis of the effect of each parameter was performed. 
The thermal load on the cornea per single pulse mainly depends on the applied radiant exposure, whereas the thermal load on the cornea for a whole treatment mainly depends on the specific pulse sequencing (i.e., on the local frequency). Even below the ablation threshold, laser pulses contribute to the thermal load. 
Radiant exposure plays the most important role in the determination of the thermal load per single pulse on the cornea. The range of radiant exposures of the excimer laser systems for refractive surgery available in the market runs from approximately 90 mJ/cm2 to approximately 500 mJ/cm2, corresponding to thermal loads of approximately 0.1 K and approximately 0.6 K, respectively. 
Figure 4 shows a linear relation between fluence and temperature load. There are other papers showing that temperature load has another slope before reaching ablation threshold. At that point, the energy distribution is changed. Before reaching the threshold, nearly all energy is used to heat tissue. Above the threshold, the main part of the energy is used as kinetic energy for the ablation particles. This would have an influence on the graph (the slope of the curve for the thermal load in Fig. 4 would change at the ablation threshold) but, for simplicity, it was not considered in our calculations. The thermal load is given by the residual heat remaining in the cornea after the flying off of the ablation products that carry away a large amount of the deposited energy. It can be determined by considering the radiant exposure distribution F(z) as a function of depth z for values F < Fth. However, we did not consider the energy dissipation that results from the ablation. 
The loss of ablation efficiency due to non-normal incidence plays a major role as well in the determination of the thermal load per single pulse outside the central part of the cornea. For an excimer laser with Gaussian radiant exposures of 300 mJ/cm2 and 1-mm diameter spot size, ablating onto a spherical cornea of a 7.77-mm radius, the thermal load decreases from 0.33 K centrally to 0.20 K at a 5-mm radial distance. The thermal load for peripheral locations decreases as well with steeper corneal curvatures. In our example, at 3 mm radially from the central cornea, the thermal load of a single pulse reduces from 0.30 K for a spherical cornea with a 9.80-mm radius to 0.26 K for a spherical cornea with a 6.44-mm radius. 
Steeper corneas suffer more from the loss of efficiency at non-normal incidence, but this drop in efficiency occurs, not only for the thermal load, but also especially for the ablation process. Since the ablation is less in the peripheral than the central cornea and this difference increases with smaller corneal radii, it affects ablation outcomes. For that reason, this loss of ablation efficiency is compensated by adding extra ablative pulses at the peripheral locations, by increasing the pulse energy at peripheral locations, or by flattening the cornea. All those measures increase the thermal load of the treatment. 
If no local frequency controls are established for the system, the thermal load of a treatment basically depends on the thermal load of a single pulse and the repetition rate of the system. For a system using a 400-mJ/cm2 peak radiant exposure, the thermal load of a single pulse is 0.4 K. If the system fires at 400 Hz without local frequency controls, the thermal load of the treatment may exceed 25 K. If the same system includes local frequency controls of 40 Hz, the thermal load of a treatment will be confined to 3 K. 
The AMARIS system offers a novel alternative. It is laser system that automatically controls the local frequency during the delivery of the sequence of pulses, thus achieving optimum refractive outcomes, visual quality, and extremely smooth surfaces and “mild“ ablations. It accomplishes this by using a defined pulse sorting, reducing the thermal impact of the ablation by arranging laser shot energy in spatial and temporal distribution and controlling pulse re-creation time versus interspot distance, to minimize the heat propagation during the ablation procedure by dynamically limiting the allowed local frequency. 48 50  
Other systems also control local frequency effect in several ways: For instance the Wavelight system (Wavelight Laser Technologies AG, Erlangen, Germany) uses the general rule “every fifth pulse can overlap a corneal location.” 12 Using our model with their specifications, we have obtained 19 K at a 1050-Hz repetition rate (i.e., 210 Hz local frequency control), 11 K at 500 Hz (i.e., a 100-Hz local frequency control), 9 K at 400 Hz (i.e., a 80-Hz local frequency control), and 5 K at 200 Hz (i.e., a 40-Hz local frequency control). 
Other systems control the mean release frequency, 51 featuring a high-speed repetition rate, as well as constant frequency per area delivery. The latter ensures a constant, effective ablation rate; by varying the delivered pulse rate to remain constant per area (at 5 Hz/mm2), the entire ablation retains consistent delivery of energy. Using our model with their (iVIS iRES; iVIS Technologies, Taranto, Italy) specifications, we have obtained 11 K at a 2 × 500-Hz repetition rate with a 5-Hz/mm2 local frequency control). 
Other systems (e.g., Kera IsoBeam; Kera Technology, Madrid, Spain) control local frequency using the simple “every second pulse may overlap a corneal location.” Using our model with their specifications, we have obtained 2 K at a 2 × 300-Hz repetition rate (i.e., 150 Hz local frequency control). 
All these approaches should result in a minimized thermal load on the cornea, fewer induced aberrations, less need for nomograms, and better postoperative visual quality. 
Considering treatment geometries, the thermal load of a treatment depends on how pulses are distributed across the treatment surface and how densely located they are. For the same amount of ablative pulses over the same treatment area, thermal load is highest during myopia treatments, followed by myopic astigmatism, mixed astigmatism, PTK, hyperopic astigmatism, and hyperopia. In myopia treatments, the pulses are most densely located at the center were the incidence is close to normal as well. In myopic astigmatism treatments, the pulses are most densely located along a central line extending toward the peripheral cornea with non-normal incidence. In PTK treatments, the pulses are evenly distributed across the corneal surface extending toward the peripheral cornea with non-normal incidence. In hyperopic astigmatism treatments, the pulses are most densely located at two symmetrical sectors at the peripheral cornea with non-normal incidence. In hyperopic treatments, pulses are most densely located in a ring at the peripheral cornea with non-normal incidence. How pulses are distributed and how densely located they are across the treatment surface, together with the non-normal incidence at the peripheral cornea, explain the results. 
With the use of the actual pulse lists generated with the AMARIS system, the thermal load of a treatment will never exceed 4 K independent of the actual repetition rate of the laser. For a −10-D spherical myopia treatment, it estimates 3.6 K, 3.5 K for a −4-D cylindrical astigmatism treatment, 2.9 K for a mixed astigmatism treatment of +3 D of sphere with −6 D of astigmatism, 2.9 K for a PTK treatment with a 75-μm depth, 2.5 K for a +7-D cylindrical astigmatism treatment, and 1.8 K for a +5 D spherical hyperopia treatment. 
The effect of water evaporation from the corneal surface is not considered in the cooling emulation. This is a limitation of this work, as various lasers employ effluent removal systems that can influence the corneal surface evaporation rate as well as the thermal changes in the environment. 
The angle of incidence on beam energy is one variable considered, but we have not considered potential changes in reflectivity since 193-nm energy can produce a huge increase in the refractive index of a layer of surface fluid (when present); this is another limitation. 
The data presented are effectively platform specific for 193 nm; however the simulation software attempts to be a valid method (i.e., platform independent). A solid state laser with a wavelength of 213 nm and a small spot size (<1 mm) is an alternative refractive surgery approach. The literature reports 52,53 similar ablation thresholds for 193- and 213-nm laser radiation, but much lower absorption coefficients for 213 nm (ablation threshold I Th = 44 mJ/cm2 at 193 nm vs. ∼50 mJ/cm2 at 213 nm and the absorption coefficients of the cornea α = 3.57 × 106 m−1 at 193 nm vs. ∼2.1 × 106 m−1 at 213 nm). Taking this into account, for laser application with the same radiant exposure at 193 or 213 nm, the temperature rise of a single pulse is approximately 60% at 213 nm of the value at 193 nm. On the other hand, the thermal relaxation time is almost three times longer at 213 nm than at 193 nm. This means that only for very short applications (<1 s) or very slow repetition rates (<4 Hz) is the cumulative thermal load at 213 nm lower than that of its 193-nm equivalent. Above these values the cumulative thermal load at 213 nm is ∼60% higher than its 193-nm equivalent. 
At the light of the lower absorption coefficients for 213 nm, it could be further hypothesized that 213-nm ablation is more efficient than 193-nm ablation. Thus, at 213 nm with approximately two thirds of the radiant exposure at 193 nm, similar spot depths and volumes are reached. Taking this into account, the temperature increase of a single pulse is approximately 40% at 213 nm of the value at 193 nm. This means that only for very short applications (<2 s) or slow repetition rates (<73 Hz) the cumulative thermal load at 213 nm is lower than its 193 nm equivalent. Above these values the cumulative thermal loads at 213 and 193 nm are approximately the same. 
Generally, denaturation temperatures are not fixed, because denaturation is a rate-limited process that depends on the heat exposure time. Cells (keratocytes and endothelial cells) are much more vulnerable than collagen. Denaturation temperatures of fibrillar collagen are, for long heat exposures over minutes, ∼65°C or larger, depending on the number of crosslinks, which varies with age. The denaturation temperature of 40° refers to collagen molecules extracted from cornea that are much more sensitive to thermal damage than is fibrillar collagen. 
These physical effects are not independent from one another; rather, they influence one another in a complex manner. For that reason, there are some performance issues when using the VLS for comprehensively analyzing or simulating laser procedures. Currently, VLS running under Windows XP SP3 installed on a computer with a single-core Intel Pentium IV HyperThreading with 3 GHz and 2 GB RAM needs approximately 0.5 seconds for processing each ablative spot (throughput, ∼2 pulses/s), which means that, for only the six treatments simulated here three times each, a total of 2 days of full computing effort were invested. The next generation of the VLS will face these performance problems by optimizing the source code for exploiting the additional performance theoretically available in multicore systems. 
We have shown that, in the presence of local frequency controls, the thermal load of the treatments do not depend on the repetition rate of the system. However, because of the presence of local frequency controls, the duration of the treatments is no longer inversely proportional to the repetition rate. The duration of the treatments is inversely proportional to the repetition rate only a slow repetition rates (<180 Hz), and stabilizes asymptotically for high repetition rates (>1500 Hz; Fig. 9). 
Figure 9.
 
Simulation of the effect of repetition rate on the duration of the treatment for different treatment types with a similar number of pulses for the AMARIS system; both axes are plotted in a logarithmic scale. Note that the duration of the treatments is inversely proportional to the repetition rate only for slow repetition rates (<180 Hz) and stabilizes asymptotically for high repetition rates (>1500 Hz).
Figure 9.
 
Simulation of the effect of repetition rate on the duration of the treatment for different treatment types with a similar number of pulses for the AMARIS system; both axes are plotted in a logarithmic scale. Note that the duration of the treatments is inversely proportional to the repetition rate only for slow repetition rates (<180 Hz) and stabilizes asymptotically for high repetition rates (>1500 Hz).
Comprehensive modeling of the local and global physical effects that occur during corneal ablation processes can be simulated using the VLS at relatively low cost and may be used for improving the quality of the results. This model can be easily generalized for any materials other than the cornea for which the absorption coefficient and the ablation threshold for the specific laser characteristics are known (e.g., PMMA). 
Today, several approaches to importing, visualizing, and analyzing highly detailed diagnostic ocular data (corneal or ocular wavefront data) are available. At the same time, several systems can be obtained to link diagnostic systems for the measurement of corneal and ocular aberrations of the eye to refractive laser platforms. These systems are state of the art with flying-spot technology, high repetition rates, fast active eye trackers, and narrow beam profiles. With these advanced technologies, such systems offer new and more advanced ablation capabilities. The improper use of ablation algorithms that overestimate or underestimate the thermal load of the treatments may result in suboptimal refractive corrections. 
Conclusions
The thermal load on the cornea due to the ablation process is an effect that should be adequately analyzed and avoided in commercial laser systems by using sophisticated algorithms that cover most of the possible variables. The improper use of a model that overestimates or underestimates the thermal load of the treatment may result in suboptimal refractive corrections. 
The model introduced in this study provides analytical expressions for the thermal load of the treatments, including cooling and heat propagation. The model incorporates several physical (local and global) effects, as well as their interdependencies in a multifactorial manner. Furthermore, due to its analytical approach, it is valid for different laser devices used in refractive surgery, as well as for any materials for which the absorption coefficient and the ablation threshold for the specific laser characteristics are known. 
The data presented are effectively platform specific; however, the simulation software attempts to be a valid method (i.e., platform independent). Of course, to run the model for the different existing or hypothetic machines, one must know the specifics of the device. 
The development of more accurate models to improve emmetropization and the correction of ocular aberrations is important. We hope that this model will be an interesting and useful contribution to refractive surgery and will take us one step closer to this goal. 
Footnotes
 Disclosure: M. Shraiki, SCHWIND eye-tech-solutions (E); S. Arba-Mosquera, SCHWIND eye-tech-solutions (E)
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Figure 1.
 
Ablative spot and thermal load as calculated by equation 3.
Figure 1.
 
Ablative spot and thermal load as calculated by equation 3.
Figure 2.
 
Temperature change of corneal tissue for 200 flat-top laser pulses illuminating the complete corneal surface with a thermal load of 0.6 K per pulse determined for different repetition rates, as calculated by equation 5.
Figure 2.
 
Temperature change of corneal tissue for 200 flat-top laser pulses illuminating the complete corneal surface with a thermal load of 0.6 K per pulse determined for different repetition rates, as calculated by equation 5.
Figure 3.
 
Spatial heat propagation across corneal tissue for a 1-mm diameter Gaussian laser pulse with a thermal load of 0.6 K per pulse determined for different propagation times, as calculated by equation 8.
Figure 3.
 
Spatial heat propagation across corneal tissue for a 1-mm diameter Gaussian laser pulse with a thermal load of 0.6 K per pulse determined for different propagation times, as calculated by equation 8.
Figure 4.
 
The thermal load of a single pulse mainly depends on the applied radiant exposure. As the radiant exposure increases, the thermal load of a single pulse increases. Even below the ablation threshold, laser pulses contribute to the thermal load.
Figure 4.
 
The thermal load of a single pulse mainly depends on the applied radiant exposure. As the radiant exposure increases, the thermal load of a single pulse increases. Even below the ablation threshold, laser pulses contribute to the thermal load.
Figure 5.
 
Reduced thermal load due to ablation efficiency as a function of the radial distance. The thermal load was calculated for different radii of curvature and for an excimer laser with Gaussian radiant exposures of 300 mJ/cm2 and 1-mm diameter spot size. Note the reduced thermal load in peripheral locations and with steeper corneal curvatures.
Figure 5.
 
Reduced thermal load due to ablation efficiency as a function of the radial distance. The thermal load was calculated for different radii of curvature and for an excimer laser with Gaussian radiant exposures of 300 mJ/cm2 and 1-mm diameter spot size. Note the reduced thermal load in peripheral locations and with steeper corneal curvatures.
Figure 6.
 
Upper limits for the thermal load for a 20-second treatment duration. The thermal load was calculated based on the repetition rate of the system (750 Hz) and based on the local frequency limit of the system (39 Hz) for an excimer laser with thermal load per pulse of 0.6 K. Notice that without local frequency controls for this system, temperature increases of up to 65 K are predicted, but the local frequency control limits the thermal load below 4 K.
Figure 6.
 
Upper limits for the thermal load for a 20-second treatment duration. The thermal load was calculated based on the repetition rate of the system (750 Hz) and based on the local frequency limit of the system (39 Hz) for an excimer laser with thermal load per pulse of 0.6 K. Notice that without local frequency controls for this system, temperature increases of up to 65 K are predicted, but the local frequency control limits the thermal load below 4 K.
Figure 7.
 
Simulation of different treatment types with durations of 20 seconds for the AMARIS system. Notice that the thermal load of all treatments remains always below the upper limit determined by the maximum local frequency for this system, temperature increases of up to 4 K are predicted, but the simulated thermal loads range between 3.6 K for myopia treatments and 1.8 K for hyperopia.
Figure 7.
 
Simulation of different treatment types with durations of 20 seconds for the AMARIS system. Notice that the thermal load of all treatments remains always below the upper limit determined by the maximum local frequency for this system, temperature increases of up to 4 K are predicted, but the simulated thermal loads range between 3.6 K for myopia treatments and 1.8 K for hyperopia.
Figure 8.
 
Simulation of the effect of the repetition rate on thermal load for different treatment types with a similar number of pulses for the AMARIS system. Note that the thermal load of all treatments at all repetition rates remains below the upper limit determined by the maximum local frequency for this system (4 K).
Figure 8.
 
Simulation of the effect of the repetition rate on thermal load for different treatment types with a similar number of pulses for the AMARIS system. Note that the thermal load of all treatments at all repetition rates remains below the upper limit determined by the maximum local frequency for this system (4 K).
Figure 9.
 
Simulation of the effect of repetition rate on the duration of the treatment for different treatment types with a similar number of pulses for the AMARIS system; both axes are plotted in a logarithmic scale. Note that the duration of the treatments is inversely proportional to the repetition rate only for slow repetition rates (<180 Hz) and stabilizes asymptotically for high repetition rates (>1500 Hz).
Figure 9.
 
Simulation of the effect of repetition rate on the duration of the treatment for different treatment types with a similar number of pulses for the AMARIS system; both axes are plotted in a logarithmic scale. Note that the duration of the treatments is inversely proportional to the repetition rate only for slow repetition rates (<180 Hz) and stabilizes asymptotically for high repetition rates (>1500 Hz).
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