Optimization Model for UV-Riboflavin Corneal Cross-linking

Abstract

Purpose.: To develop a theoretical model for riboflavin ultraviolet-A cross-linking treatment that can predict the increase in stiffness of the corneal tissue as a function of the ultraviolet intensity and riboflavin concentration distribution, as well as the treatment time.

Methods.: A theoretical model for calculating the increase in corneal cross-linking (polymerization rate) was derived using Fick's second law of diffusion, Lambert-Beer's law of light absorption, and a photopolymerization rate equation. Stress–strain experiments to determine Young's modulus at 5% strain were performed on 43 sets of paired porcine corneal strips at different intensities (3–7 mW/cm²) and different riboflavin concentrations (0.0%–0.5%). The experimental results for Young's modulus increase were correlated with the simulated polymerization increase to determine a relationship between the model and the experimental data.

Results.: This model allows the calculation of the one-dimensional spatial and temporal intensity and concentration distribution. The total absorbed radiant exposure, defined by intensity, concentration distribution, and treatment time, shows a linear correlation with the measured stiffness increase from which a threshold value of 1.7 J/cm² can be determined. The relative stiffness increase shows a linear correlation with the theoretical polymer increase per depth of tissue, as calculated by the model.

Conclusions.: This theoretical model predicts the spatial distribution of increased stiffness by corneal cross-linking and, as such, can be used to customize treatment, according to the patient's corneal thickness and medical indication.

Ultraviolet (UV) cross-linking increases the biochemical stability of stromal tissue and, as such, has become an established method of treating progressive keratoconus and other corneal ectasias. ^{1 –12} Additional chemical bonds are created during cross-linking inside the corneal stroma, most probably intrafibrillar. ¹³  

A standard protocol has been established on the basis of the initial work performed by Spoerl et al. ¹⁴ The first step involves abrasion of the corneal epithelium and administration of a 0.1% riboflavin solution to the cornea for 30 minutes (1 drop every 3 minutes). This method ensures the desired distribution of the riboflavin throughout the stroma before illumination with ultraviolet light at a wavelength of 370 nm (the absorption maximum of the riboflavin chromophore). During the illumination phase, additional riboflavin is administered (every 5 minutes) to compensate for its ongoing consumption. The riboflavin solution acts as a photosensitizer that creates free radicals which, on irradiation, lead to the formation of new chemical bonds. ^10,14 The irradiation intensity used for the standard protocol is 3 mW/cm² for a period of 30 minutes, which corresponds to a radiant exposure of 5.4 J/cm² at the anterior surface of the cornea. 

These standard parameters were established with two aspects in mind: first, safety, to protect the endothelium and other posterior structures of the eye, and second, effectivity, to compensate the reduced stiffness in keratoconic corneas. To the best of our knowledge, there are no reports available that have investigated the possibility of achieving a maximized increase in biomechanical strength. Therefore, different sets of parameters might be equally safe and could perhaps even better address specific medical indications. Our purpose is thus to construct a model that will predict the distribution of generated cross-links within the corneal stroma and thus the increase in Young's modulus. The model considers the riboflavin concentration, the pretreatment time, the light intensity, the illumination time, and the individual thickness of the patient's cornea. The model presented also describes the spatial and temporal increase in Young's modulus. In this case, we assume that an increase in corneal stiffness is related to an increase in new chemical bonds, which will be mathematically described by a photopolymerization process. Photopolymerization is a result of interaction between proteins and reactive singular-oxygen created during the UV-riboflavin cross-linking treatment. 

Methods

Theoretical Model

There are five principle steps of the modeling process (calculate diffusion of the photosensitizer, calculate light distribution, solve the rate equation of photopolymerization, determine the grade of polymerization, and calculate the relative stiffness increase). The polymerization process induced by the UV cross-linking procedure is described by a rate equation that requires the spatial and temporal concentration distribution of the photosensitizer and the distribution of the illuminating light in the tissue. To derive a stiffness increase from a grade of polymerization, it is necessary to have the support of experimentally obtained data to correlate the relative increase in corneal stiffness with the calculated increase in cross-links. 

The following approximations are made to allow an analytical solution of the rate equation based on the concentration and light distribution within the stroma: First, the cornea is regarded as a homogeneous tissue. This assumption reduces the model to one dimension, going from anterior to posterior. Second, an unlimited reservoir of the photosensitizer solution is available on the corneal surface and the consumption of the riboflavin during the illumination phase is equal to the amount of riboflavin that is readministered, meaning that the concentration gradient within the stroma does not change during the illumination time. Third, the absorption of the riboflavin layer on top of the corneal surface is ignored during illumination. Fourth, light scattering within the tissue is ignored. 

Diffusion Model

The diffusion of the riboflavin solution into the corneal stroma can be described by Fick's second law of diffusion:  

 

where c(z,t) is the photosensitizer (riboflavin) concentration, D the diffusion coefficient, z the depth in the tissue, and t the diffusion time (pretreatment time). 

The following boundary conditions apply to the diffusion of the riboflavin into the stroma: First, before the application of the photosensitizer, no photosensitizer solution should be present in the corneal tissue. Second, the repeated application of the photosensitizer solution at the corneal surface should result in a constant thin film of the photosensitizer solution so that, at all times, a constant concentration, c ₀, is available at the corneal surface. Third, at an infinite distance from the corneal surface, the concentration should be 0. 

 

 

 

A suitable solution of the differential equation for the diffusion (equation 1), applying the boundary conditions, is equation 2, the complementary error function, erfc(z). 

 

 

 

Using equation 3, the photosensitizer concentration, c(z,t), within the cornea can be calculated both spatially and in a time-dependent fashion for a given diffusion coefficient, D. 

Light Distribution and Absorption

The light distribution inside the cornea, I(z,t), can be derived from Lambert-Beer's Law when, in a first approximation, light scattering in the tissue is ignored. 

 

 

 

Here, I ₀ is the intensity at the corneal surface, μ_t, the total absorption coefficient; and z, the distance from the surface. The total absorption can be split into the absorption by the corneal stroma, μ_c, and the absorption by the photosensitizer (e.g., riboflavin), μ_p (μ_t = μ_c + μ_p). The absorption by the photosensitizer, riboflavin, depends on the molar extinction coefficient, ε_r, and the concentration c(z,t). 

Equation 4 for the light distribution can thus be rewritten as  

 

 

In equation 5, the spatial- and time-dependent intensity distribution, I(z,t), can be calculated using the following analytical solution of the integral over the concentration:  

 

where erf is the error function. 

Polymerization Model

Polymerization is a process in which monomer molecules bind together in a chemical reaction to form three-dimensional networks or polymer chains. Riboflavin-initiated polymerization is based on chemical chain propagation, a type of polymerization in which the reactive center of a polymer chain consists of a radical. The free radical reaction mechanism can be divided into three stages: initiation, chain propagation, and chain termination. ^15,16  

Thus, the rate of polymerization, d[M]/dt, can be described by  

 

where R _i is the rate of initiation, R _p the rate of chain propagation, and R _t the rate of chain termination. 

One essential assumption in deriving the rate of polymerization is that the rate of reacting monomers, R_i, and terminated monomers, R_t, is significantly smaller than R_p, and, as an approximation, both can be neglected. Thus, the spatial- and time-dependent rate of polymerization of the corneal tissue can be rewritten as:  

 

where R₀ is a constant that combines several unknown material-specific constants. The detailed derivation of this equation is shown in Appendix A. Consequently, the total amount of induced cross-links, M(z,t), can be obtained by integrating equation 8 over time and space:  

 

 

In addition, the total amount of absorbed energy, E _eff, within the tissue can be derived by:  

 

where z _max is the maximum depth of the tissue. 

Stress–Strain Experiments

The purpose of the stress–strain measurements is to compare the theoretically calculated polymer generation during cross-linking to an experimentally measured increase in corneal stiffness. 

The stress–strain measurements use fresh porcine corneas, a widely accepted model of the human cornea. ^11,12 Before corneal cross-linking (CXL) treatment, the epithelium was abraded, and the thickness of the cornea was determined (Pachymeter SP-2000; Tomey, Nagoya, Japan). Subsequently, two small slices of cornea measuring 9 × 1 mm were dissected with a custom-made, three-blade knife. The upper and lower 1-mm ends of the slice were used for fixing the slice in the sample grips of the stress–strain material-testing machine (MPM 145670; Zwick-Roell, Ulm, Germany). The defined 7:1-mm proportion of the slice is important because it suppresses shear forces caused by the material geometry during the stress–strain measurement. The dissected slices were placed in dextran solution with different riboflavin concentrations (0.0%, 0.05%, 0.1%, 0.125%, 0.3%, or 0.5%) for at least 18 hours. The long pretreatment time ensures that the riboflavin concentration is homogeneously distributed throughout the tissue, and no concentration gradient exists. One of the two slices was used as a control and was not cross-linked. 

Two series of measurements were performed. In the first series, the irradiation intensity of the illumination system (UV-X 1000; IROC, Zurich, Switzerland) was kept constant at 3 mW/cm² for 30 minutes, and the concentration of riboflavin was increased from 0.0% to 0.5%. In a second series, the riboflavin concentration was kept constant at 0.125% and the irradiation intensity of the illumination system was increased from 3 to 7 mW/cm². Table 1 shows an overview of the parameters used. 

Table 1.

 

Parameters of the Corneal Probes Used for the Stress–Strain Measurements

Series	Group	Number of Valid Measurements	Riboflavin Concentration (%)	Irradiation Intensity (mW/cm2)	Treatment Time (min)	Dosis (J/cm2)
1	1	2	0	3	30	5.4
1	2	6	0.05	3	30	5.4
1	3	8	0.1	3	30	5.4
1	4	4	0.3	3	30	5.4
1	5	7	0.5	3	30	5.4
1/2	6	6	0.125	3	30	5.4
2	7	7	0.125	6	30	10.8
2	8	3	0.125	7	30	12.6

Before the stress–strain measurement, one cornea slice was cross-linked and the other was kept in the dark by covering it with a black box. Dehydration was avoided by administering dextran solution every 5 minutes to both samples. After the UV cross-linking on the one sample, stress–strain measurements were performed in a random sequence of the two samples, according to the same protocol. 

The sample was placed in the apparatus and prestretched seven times with a lower amount of displacement (1%–5% of strain) to compensate for the viscoelasticity of biological tissue. In the eighth turn, which achieves a 12% strain, the force, F, is measured. From this, the stress, σ, can be determined by dividing the force, F, by the cross-section, A, of the sample. Knowing the total strain, ε, an approximation of Young's modulus, E, can be determined directly from the proportionality:  

 

 

The relative increase in stiffness, rel.ΔE, of each cornea can be obtained by comparing the UV-cross-linked sample with the untreated sample:  

 

 

Merging Model and Experiment

The increase in stiffness as a function of the generated UV cross-linked polymers can be obtained by merging the model and the stress–strain measurements. Thus, the relative increase in polymers per depth, rel.ΔM, has to be determined by dividing the total number of generated polymers, [ΔM], by the corneal thickness z ₀. 

 

 

 

Furthermore, the model is adapted to describe the experimental conditions rather than an in vivo human cornea (e.g., porcine cornea, homogeneous riboflavin concentration c(z,t) = c ₀). 

Results

Theoretical Results of the Model

The spatial distribution of the photosensitizer (riboflavin) concentration as a function of the diffusion time (pretreatment time), t, calculated with equation 3 is shown in Figure 1A. A standard riboflavin concentration of 0.1% was applied to the surface of the cornea, and a literature-based ¹⁷ diffusion coefficient of D = 6.5 × 10⁻⁷ cm²/s was used. 

Figure 1.

 

Concentration of the photosensitizer riboflavin (A), the light distribution (B), and the rate of polymerization (C) in the corneal tissue as a function of the corneal depth for different pretreatment times. The concentration at the corneal surface is 0.1%, the intensity at the corneal surface 3 mW/cm² and the parameter R ₀ = 1.

The derived distribution of the photosensitizer molecules permits the calculation of the light distribution in the corneal tissue as a function of the pretreatment and diffusion times, respectively. Figure 1B shows the obtained light intensity as a function of the diffusion time of the photosensitizer molecules and the depth of the cornea using equation 5. Here, a light intensity at the surface of the cornea is considered to be I(0,t) = 3 mW/cm². The molar extinction coefficient ε_r = 10,066 L (cm · mol; for 365 nm) was taken from the literature ¹⁸ and an absorption coefficient of the cornea, μ_c = 17 μm⁻¹, (own measurements) was used. Figure 1B demonstrates that, with increasing pretreatment time, more photosensitizer molecules diffused into the tissue, causing a decrease in light intensity in the posterior of the cornea. 

The rate of generated polymers owing to the time-dependent distribution of the riboflavin molecules and the light, as calculated by equation 8, is shown in Figure 1C. For short pretreatment times (<5 minutes), the rate of polymerization was almost 0 in the posterior of the cornea. This observation is explained by the fact that the short diffusion time precludes molecules from generating cross-links. With increasing pretreatment time, the polymerization rate increases as more molecules become available. A higher concentration of the photosensitizer molecules in the anterior of the cornea may prevent the UV light from penetrating into deeper structures of the cornea and, as a result, may also limit a further increase in the polymerization rate or could even decrease the polymerization rate in the posterior of the cornea. 

Experimental Results

The increase in corneal stiffness, which is later associated with the increase in polymers per depth of tissue can be determined experimentally. Figure 2 shows the stiffness increase as a function of the calculated total absorbed radiant exposure (equation). The increase in stiffness correlates significantly with an increase in the total absorbed radiant exposure. One can estimate the threshold radiant exposure necessary to achieve an increase in stiffness of 1.7 J/cm² with a 95% CI ranging from 0.3 to 2.5 J/cm². Thus, a minimum amount of energy must be absorbed to achieve a significant increase in stiffening. The negative values of relative stiffness increase are questionable, but represent the limitation of such stress–strain measurements with viscoelastic materials, such as corneal tissue. 

Figure 2.

 

Relative stiffness as a function of the total absorbed energy for different riboflavin concentrations and 3 mW/cm², as well as different irradiation intensities and constant riboflavin concentration of 0.1%. The total absorbed radiant exposure was calculated using equation 10.

The experimental parameters used for the corneal strips were applied to the model, and the resulting polymers per depth of tissue were calculated as a function of these parameters. Figure 3 shows the relationship of the data from the model (polymers per micrometer tissue depth) and the experiments (increase in relative stiffness). Assuming a linear relationship for this range of parameters, the relative stiffness increase, rel.ΔE, predicted by the model can be described to a good approximation by the following equation:  

 

 

Figure 3.

 

Relationship between polymer increase per depth of corneal tissue (obtained from the model) and relative stiffness increase (obtained by the experiments) for different riboflavin concentrations and 3 mW/cm² as well as different irradiation intensity and constant riboflavin concentration of 0.1%.

Discussion

The model presented here describes a theoretical approximation of the CXL procedure. It predicts an optimal riboflavin concentration for maximizing the increase in corneal stiffness and can be used to determine the optimal treatment parameters (concentration, light intensity, and pretreatment and treatment times) for a strong surface cross-linking effect or volume effect. In addition, the model predicts a nonlinearity of the cross-linking effect. 

The maximum corneal stiffness increase can be predicted not only for a thin slice of cornea but also for the entire human cornea by applying the determined linear relationship between polymer increase per depth of corneal tissue and the increase in relative stiffness compared with the whole cornea. This, however, assumes that the correlation factor of 3.6 is also valid for different sets of parameters. Figure 4 shows the increase in stiffness as a function of the riboflavin concentration (Fig. 4A) and irradiation intensity (Fig. 4B) of this simulation for a corneal thickness of 550 μm. Up to a concentration of 0.15%, an increasing riboflavin concentration results in a higher stiffness increase (cf. Fig. 4A). However, if the riboflavin concentration is increased even further, the stiffness increase is reduced because the riboflavin blocks the irradiation light, which then cannot reach the deeper layers of the cornea. Specifically, the more uneven the distribution of cross-links generated over the thickness of the cornea, the lower the total stiffness increase. Furthermore, the simulation shows that, with increasing intensity, the total stiffness increase achieved can be maximized (Fig. 4B). However, because the radiation could reach the endothelium and cause damage, the intensity cannot be increased too high. Nevertheless, the intensity limitation was not implemented in this model. 

Figure 4.

 

Relative stiffness increase as a function of (A) the riboflavin concentration at different irradiation intensities. Irradiation time was 30 minutes and (B) the intensity at different riboflavin concentrations.

The optimum stiffness increase determined by the model (Fig. 4) can be explained by the fact that, if the initial riboflavin concentration at the surface is relatively high, the irradiation light is blocked by the riboflavin and thus no cross-links are generated in the deeper layers of the cornea. Of course, more cross-links are generated closer to the surface, but this only increases the stiffness in the upper layers. However, in stress–strain measurements, the total increase over the total depth of the cornea is measured and, therefore, a distribution with a steep gradient between the anterior and the posterior of the cornea would not seem to be ideal. In the stress–strain experiments, it is important that as many cross-links as possible are generated and, preferably, evenly distributed. Even so, this may contradict the natural biomechanical behavior of the cornea as the anterior stroma typically has a larger Young's modulus than the posterior cornea. 

These considerations can also be used to illustrate the need for different treatment parameters for surface and volume cross-linking applications: A thin layer of cross-linked tissue at the surface is achieved with a high riboflavin concentration, whereas a large volume of cross-linked tissue is achieved only with a low concentration and longer pretreatment times. 

Kohlhaas et al. ¹² have studied the increase in Young's modulus in three different layers of pig corneas using 3 mW/cm² for 30 minutes (5.4 J/cm²) and a concentration of 0.1% riboflavin solution. The relative increase in Young's modulus in the first 200 μm of the cornea was measured to be 6.3 MPa, whereas the values for the midstroma (200–400 μm) were found to be 2.7 MPa at 5% strain. The untreated control groups showed a Young's modulus of 2.9 MPa for the anterior segment and 2.8 MPa for the intermediate section. Thus, the relative stiffness increase was 0 in the midsection and 33% in the anterior. Our model reflects such measurements and would predict an increase in stiffness by 55% (anterior stroma) and 24% (mid-stroma), similarly demonstrating an uneven distribution of the biomechanical effect. The increase our model predicts is larger than the measurements by Kohlhaas et al., ¹² which, on the one hand, can be explained by the limitations of the model discussed below, and on the other hand, by the limitation of the experimental measurements of Young's modulus in different stromal layers. 

Nevertheless, different distributions of the biomechanical stability may be of relevance for different clinical conditions: In thin cornea (e.g., post LASIK or keratoconus), one might prefer a stronger stiffness layer in the anterior part of the cornea compared with cornea showing a more normal corneal thickness (e.g., early stage of keratoconus), where one may require a more even biomechanical increase throughout the entire corneal thickness, However, possible damage thresholds of the endothelium must always be kept in mind. Finally, thick cornea (e.g., in Fuch's dystrophy) may require parameters that allow an even distribution, even at corneal depths of 600 to 700 μm. 

Theoretically, this model can help to optimize CXL treatment in several ways for different corneal thicknesses (thickness of cross-linking layers). It can be optimized for treatment time, maximum stiffness increase, and homogeneous stiffness throughout the corneal thickness. The treatment parameters might be optimized individually, in consideration of the individual patient's corneal thickness. 

For this model, it was assumed that the irradiation intensity and therefore the induced polymerization rate by singular oxygen could theoretically be increased without any limit, and thereby neglecting the fact that endothelial damage can occur if the irradiation intensity is too high. Using cell cultures, Wollensak et al. ¹⁹ showed that UV irradiation without application of riboflavin as a photosensitizer causes endothelial cell damage at irradiation levels of 4 mW/cm² for 30 minutes (radiant exposure, 7.2 J/cm²). With the additional application of riboflavin (concentration of 0.025%), the damage threshold is lowered by a factor of 10, to an irradiance of 0.35 mW/cm² or a threshold radiant exposure of 0.63 J/cm². This raises the question of whether the damage is related to the applied intensity or the induced radicals surrounding the endothelial cells. According to our model, the induced radicals are related to the combination of intensity, concentration, and time (equation 8) by the correlation coefficient R ₀. Remodelling the data published by Wollensak et al. ¹⁹ in consideration of their cell culture setup, we derived a damage threshold value for the endothelium cell apoptosis of d[M]/dt = 0.003 × R ₀ for an induced rate of radicals (cross-links). Similar values (d[M]/dt = 0.002 × R ₀) for the damage threshold rate of induced radicals can be derived by remodelling the data of the rabbit study (corneal thickness, 400 μm) published by the same research group. ²⁰ Furthermore, recalculating the keratocyte cell damage in different depths of keratocyte loss, the threshold value varies between d[M]/dt = 0.003 × R ₀ and d[M]/dt = 0.006 × R ₀. ²¹ Thus, the damage threshold of the induced polymerization rate for cell structures seems to belong to a specific rate of induced radicals (cross-links). In our current model, this value tends toward d[M]/dt = 0.003 × R ₀. The consequences of modeling CXL are demonstrated in Figure 5. Modifying intensity, concentration distribution, or treatment time would result in different cross-link depth with a different amount of induced cross-links and therefore a different increase in corneal stiffness. A consequence of these results is that CXL does not follow the Bunsen-Roscoe law of reciprocity. ²² (A specific biological effect is directly proportional to the total energy dose,, regardless of the regimen.) Our results show that cross-linking is not directly proportional to the total irradiation dose. This initial assumption was used to explain the equivalence of experimental results concerning the stiffness increase of porcine corneal strips that were treated with a rapid (higher intensity and shorter treatment time) or standard CXL procedure. ²³ Similar results could be shown by other studies that applied a constant energy dose of 5.4 J/cm². Roizenblatt et al. (IOVS 2010;51:ARVO E-Abstract4979) showed a statistically equivalent increase in corneal stiffness after cross-linking, using an irradiation of 3 mW/cm² for 30 minutes and of 9 mW/cm² for 10 minutes. A study by Krueger et al. ²⁴ showed equivalence for groups illuminated with 2 mW/cm² for 45 minutes, 3 mW/cm² for 30 minutes, 9 mW/cm² for 10 minutes, and 15 mW/cm² for 6 minutes. Although these results validate the Bunson-Roscoe law, there is evidence against its validity, published by Lanchares et al. ²⁵ In this study, rabbit corneas were illuminated with 3 mW/cm² for 30 minutes and with the same intensity for 60 minutes. As expected, a statistically significant increase in corneal stiffness was found for the standard parameters of 3 mW/cm² for 30 minutes. However, no increase was found for the corneas that have been illuminated for 60 minutes. Assuming the validity of the Bunson-Roscoe law, in the 60-minute group, the stiffness increase should be larger than in the standard group, as a consequence of the increased energy dose. Thus, the evidence on whether or not the Bunson-Roscoe law can be directly applied to CXL is inconclusive. 

Figure 5.

 

Concentration distribution, light intensity, rate of induced cross-links, and increase in stiffness as a function of the corneal depth. Four different cases are shown: c ₀ = 0.1%, treatment time 30 minutes, and intensity 3 mW/cm²; c ₀ = 0.1%, treatment time 9 minutes, and intensity 10 mW/cm²; c ₀ = 0.05%, treatment time 30 minutes and intensity 3 mW/cm²; and c ₀ = 0.05%, treatment time 9 minutes, and intensity 10 mW/cm². The pretreatment time was set to 30 minutes. Dotted lines: the threshold for the onset of induced cross-links.

In the derived model, the rate of cross-links depends on the square root of the intensity. Thus, the increase in stiffness is not the same if two different products of intensity and time, which equal the same energy dose, are used, as shown in Figure 5 (compare light and dark blue or red and green). Higher intensities for a shorter treatment time result in a decrease in the amount of total cross-links by a factor of 0.7 to 0.8 combined with an increase in cross-linking depth by a factor of 1.4 to 1.6. This increase in cross-linking depth might compensate for the decrease in total amount of cross-links and supports the assumption (as does equivalence reported from experimental investigations) that, not only the total number of cross-links, but also the distribution of the induced cross-links, contributes to the effective increase in stiffness. ²³  

This presented model shows similar behavior when the physical and biological parameters are changed. As we are currently using values from the literature or from our own experimental results, it is worth investigating the influence of individual parameters on the simulations. The diffusion coefficient was found to have the least influence, as the calculated polymerization rate varies only slightly. The increase in the corneal light absorption exerts a larger influence, especially at lower riboflavin concentrations. Indeed, although absorbance = absorption + scattering, in this model, as well as in all other investigations of CXL, absorption equals the absorbance as the influence of the scattering is neglected. However, for a wavelength of 365 nm light, scattering exerts a large influence on the light distribution inside the tissue ²⁶ and must be considered if a model with a higher level of detail is desired. 

The most relevant parameter is the absorption of the riboflavin, which is dependent on the extinction coefficient (constant) and the concentration. If the calculation of concentration distribution is varied, not only the total amount of cross-linking is changed, but also the optimum initial concentration administered at the corneal surface. The calculated concentration is the parameter, which is most influenced by the simplifications made by deriving the model. A consequence of the modeled concentration distribution can be observed by recalculating the threshold values for endothelial cell damage published by Wollensak et al. ²⁰ in 2003. As the authors themselves realized, ¹⁹ the intensity at the endothelium is higher than initially calculated. Therefore, the threshold values of 0.36 mW/cm² (0.65 J/cm²) must be corrected by a factor of approximately 2 or 3. 

It is worth mentioning that this model still neglects the limited pool of riboflavin and thus idealizes the photosensitizer concentration in the corneal tissue. To better understand the concentration distribution, online monitoring of the spatial distribution would be advantageous. Each drop of riboflavin that is applied to the cornea offers only a certain number of riboflavin molecules that can diffuse into the corneal tissue. At the moment, we assume that, due to constant application of additional riboflavin drops, the pool at the corneal surface is unlimited, and thus the average amount of riboflavin corresponds to the initial applied concentration. Estimations show that the average concentration within an interval (5 minutes) between two riboflavin drops is instead only 40% to 60% of the initial riboflavin concentration. ²⁷ Therefore, considerably less riboflavin than what is assumed in the model is actually entering the corneal tissue. Furthermore, at this stage, the model postulates that the amount of riboflavin consumed by the photochemical reaction is refilled exactly by the same amount, owing to the regular application of more riboflavin during the irradiation time. This assumption has unknown effects on the results. In addition to the simplification caused by the riboflavin application, the unidimensionality of the model does not take into account the changing spatial thickness of the cornea and its curvature. This results in neglecting possible uneven light distributions caused by a gradient light source, higher reflection losses, and a larger projected area in the periphery. Finally, the absorbing riboflavin layer on top of the cornea, which will block parts of the initial UV light is likewise neglected. 

Despite these limitations, this model, the first of its kind presented for optimization of UV CXL treatment, shows that the commonly used standard procedure, which does not account for any individual conditions of the patient's cornea, can be optimized. It is possible to calculate the amount of generated polymers in the corneal tissue and predict the relative stiffness increase. At this stage the model is scaled to experimental data, in which the corneal stiffness over the full corneal depth is averaged. As a consequence, the predicted results should be considered qualitative. Depth-dependent corneal tensile strength experiments would further validate the performance of the model. 

By adding further details and more precise, depth-dependent data to a future model, it is conceivable that individualized custom treatments the are well tailored to the patient's condition can be planned and will provide a better outcome of the treatment than that which is currently achievable using the standard procedure. 

Footnotes

 Supported by IROC AG, internal research budget. Authors' appointments and work were carried out at the IROC AG, Institute of Refractive and Ophthalmic Surgery, Zurich, Switzerland.

Footnotes

 Disclosure: S. Schumacher, IROC AG (E); M. Mrochen, IROC AG (E); J. Wernli, IROC AG (E); M. Bueeler, None; T. Seiler, IROC AG (E)

Appendix A

Polymerization Model

The free radical reaction mechanism can be divided into three stages: initiation, chain propagation, and chain termination. Initiation is the creation of free radicals necessary for propagation. The radicals can be created from radical initiators, such as riboflavin under UV light exposure. The products formed are unstable and easily break down into two radicals. The radical then is transferred to the first monomer molecule to form the radical monomer molecule, M ₁·, which initiates the chain reaction. The rate of initiation is defined as R _i. 

Propagation is the rapid reaction of this radicalized molecule with another monomer, and the subsequent repetition to create a repeating chain. The rate of propagation is defined as R _p. 

Termination occurs when a radical reacts in a way that prevents further propagation. The most common method of termination is by coupling, where two radical species react with each other to form a single molecule. Another, less common, method of termination is chain disproportionation where two radicals meet but, instead of coupling, exchange a proton, and two terminated chains ensue: one saturated and the other with a terminal double bond. 

The rate of polymerization, d[M]/dt, can be described by using the monomer degradation rate:  

 

where R _i is the initiation rate, R _p is the propagation rate, and R _t the termination rate. At first, it is assumed that the rate of reacting monomers, R _i, and terminated monomers, R _t, are significantly smaller than R _p (see below) and, as a first approximation, both can be neglected. 

 

 

 

The propagation rate, R _p, and therefore the polymerization rate, represents the summation of propagation steps, where the propagation rate is equal to all steps. The equation (equation A2) thus can be written as follows:  

 

where [M] is the monomer concentration, [M·] is the total concentration of all chain radicals, and k _p is the polymerization velocity coefficient. Unfortunately, equation A3 is not very helpful, because the concentration of the chain radical, [M·], is very low, extremely difficult to measure, and thus unknown. To eliminate [M·], one can make the assumption that, after an initial increase, the concentration of the radicals is constant during the propagation phase. This assumption is equivalent to the statement that the rate of initiation, R _i, and termination, R _t, compensate each other and thus cancel out. 

 

 

 

The right side of the equation represents the rate of termination, R _t, and k _t is the termination velocity coefficient. Thus, equation A3 can be rewritten as:  

 

 

When equation A5 is inserted into equation A3, the equation for the polymerization rate becomes  

 

 

Furthermore, the rate of initiation, R_i, at a certain depth in the tissue is dependent on the properties of the photoinitiator (riboflavin) and the amount of light and is defined as follows:  

 

where Φ is the efficiency of the photoinitiator in forming radicals, and I_a is the amount of light absorbed by the photoinitiator per unit of depth, d, at distance z. Considering that d is infinitesimally small, I_a an be regarded as:  

 

 

The differential notation of the Lambert-Beer law provides the following expression:  

 

and unites the diffusion, the light distribution, and the polymerization model. By reverse substitution of the equations, one obtains the following expression for the rate of polymerization:  

 

 

This is a first-order differential equation where the change of [M] with time is dependent on [M]. However, considering that the riboflavin at the corneal surface is constantly readministered, [M] can be considered as being inexhaustible and, therefore, in a first approximation, as constant. Thus, the constants k _p, k _t, Φ, ε_r, and [M] can be summed up into one constant, R ₀. As such, equation A11 reads: