**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/cm^{2}) 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/cm^{2} 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.

^{ 1 –12 }Additional chemical bonds are created during cross-linking inside the corneal stroma, most probably intrafibrillar.

^{ 13 }

^{ 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/cm

^{2}for a period of 30 minutes, which corresponds to a radiant exposure of 5.4 J/cm

^{2}at the anterior surface of the cornea.

*c*

_{0}, is available at the corneal surface. Third, at an infinite distance from the corneal surface, the concentration should be 0.

*erfc*(

*z*).

*c*(

*z*,

*t*), within the cornea can be calculated both spatially and in a time-dependent fashion for a given diffusion coefficient,

*D*.

*I*(

*z*,

*t*), can be derived from Lambert-Beer's Law when, in a first approximation, light scattering in the tissue is ignored.

*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*).

*I*(

*z*,

*t*), can be calculated using the following analytical solution of the integral over the concentration: where

*erf*is the error function.

^{ 15,16 }

*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

*R*

_{0}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:

^{ 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.

^{2}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/cm

^{2}. Table 1 shows an overview of the parameters used.

Series | Group | Number of Valid Measurements | Riboflavin Concentration (%) | Irradiation Intensity (mW/cm^{2}) | Treatment Time (min) | Dosis (J/cm^{2}) |
---|---|---|---|---|---|---|

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 |

*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:

*M*, has to be determined by dividing the total number of generated polymers, [Δ

*M*], by the corneal thickness

*z*

_{0}.

*c*(

*z*,

*t*) =

*c*

_{0}).

*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}cm

^{2}/s was used.

*I*(0,

*t*) = 3 mW/cm

^{2}. 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.

^{2}with a 95% CI ranging from 0.3 to 2.5 J/cm

^{2}. 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.

*E*, predicted by the model can be described to a good approximation by the following equation:

^{ 12 }have studied the increase in Young's modulus in three different layers of pig corneas using 3 mW/cm

^{2}for 30 minutes (5.4 J/cm

^{2}) 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.

^{ 19 }showed that UV irradiation without application of riboflavin as a photosensitizer causes endothelial cell damage at irradiation levels of 4 mW/cm

^{2}for 30 minutes (radiant exposure, 7.2 J/cm

^{2}). 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/cm

^{2}or a threshold radiant exposure of 0.63 J/cm

^{2}. 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/cm

^{2}. 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/cm

^{2}for 30 minutes and of 9 mW/cm

^{2}for 10 minutes. A study by Krueger et al.

^{ 24 }showed equivalence for groups illuminated with 2 mW/cm

^{2}for 45 minutes, 3 mW/cm

^{2}for 30 minutes, 9 mW/cm

^{2}for 10 minutes, and 15 mW/cm

^{2}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/cm

^{2}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/cm

^{2}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.

^{ 23 }

^{ 26 }and must be considered if a model with a higher level of detail is desired.

^{ 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/cm

^{2}(0.65 J/cm

^{2}) must be corrected by a factor of approximately 2 or 3.

^{ 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.

*M*

_{1}·, which initiates the chain reaction. The rate of initiation is defined as

*R*

_{i}.

*R*

_{p}.

*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.

*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.

*R*

_{t}, and

*k*

_{t}is the termination velocity coefficient. Thus, equation A3 can be rewritten as:

*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:

*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: