The recent article of Bekesi et al.

^{1}made two important conclusions about corneal collagen crosslinking (CXL): Firstly, biomechanical properties determined after ex vivo CXL may not provide entirely accurate information about the responses to CXL in vivo; secondly, within the crosslinked regions, rose bengal crosslinking (RGX) stiffened the cornea to a greater degree than ultraviolet-A (UVA) crosslinking (UVX) at 1 and 2 months, suggesting a greater density of crosslinks produced by RGX. The intent of this letter is to provide mathematical formulas to analyze and comment on the above clinically measured features and also propose a new protocol called the riboflavin concentration-controlled method (CCM), which, as theoretically demonstrated, is more efficient than the conventional Dresden protocol (using a saturated concentration) or other noncontrolled concentration methods based on the Bunsen–Roscoe reciprocal law.For type I, epi-off CXL, the efficacy is given byDisplay Formula \({\rm{S}} = \sqrt {(4{{kF\rm Co}}/({{aX\rm Io}}){\rm{\ }}} {\rm{\ }}[1 - \exp \left( { - {\it{bX}}} \right){]}\), where Display Formula \({{\rm{S}}^{\rm{*}}} = 0.65\sqrt {(4{k\rm{Co}}/({{a}}{{{X}}^{\rm{*}}}{\rm{Io}}){\rm{\ }}}\), which follows a nonlinear scaling lawDisplay Formula \({{\rm{I}}^{\rm{*}}} = 1.7{\rm{kFCo}}/({\rm{aXS^{\prime 2}}})\), which is approximately 45 to 60 mW/cm

^{2,3}Ceff = 1 − exp(−S) with S being the crosslinking rate function given by*k*is an effective rate constant;*X*= exp(−*A*z); z is the stromal thickness;*A*is an effective absorption coefficient given by*A*= 2.3*ma*(1 − 0.25z/_{2}C_{0}*D*) + Q;*b*= 0.5*a*(tI_{0}),*a*= 0.622*p*,*p*being the quantum yield and*m*= 1.5 a fit parameter;*a*_{2}and*Q*are the absorption coefficient of the photolysis and the stroma, respectively. C_{0}is the initial (at t = 0) riboflavin (RF) concentration having a diffusion function defined by a diffusion depth (*D*),*F*(z) = 1 − 0.5z/*D*; and I_{0}is the initial UV light intensity on the corneal surface (at z = 0). The peak value of S (when*bX** = 1.25, with a mean value*F*= 0.75) is given by^{3}that S* is proportional to (E_{0}/I_{0})^{0.5}, or t^{0.5}, in contrast to the Bunsen–Roscoe law (BRL) that S is proportional to E_{0}such that the irradiation time in accelerated CXL is given by a reciprocal law t = E_{0}/I_{0}. Our S-formula shows that, for the same dose, high UV powers (9–40 mW) deplete the RF faster and reach a lower steady-state efficacy and shallower crosslink depth than the conventional lower power (1.5–3 mW). For a given efficacy threshold (S'), the S-formula may be used to calculate the maximum (or cutoff) intensity^{2}, and the associated minimum crosslink time (T* = E_{0}/I*), approximately 2 to 3 minutes, which provide the valid range of BRL and have been demonstrated by the clinical data of Wernli et al.^{4}For example, for CDisplay Formula \(\sqrt {(4{kF\rm{Co}}/({aX\rm{Io}}){\rm{\ }}} {\rm{\ }}\), showing that the anterior stroma always has lower efficacy than the posterior, whereas in type II CXL strength is proportional toDisplay Formula \(({akF\rm{CoXIo}}[{O_2}])\) which has a reverse dependence on z, because

_{0}= 0.1%,*a*_{2}= 50 (1/(cm%)),*k*= 0.39(1/s), and quantum yield*p*= 0.5, we obtain steady-state S' = 14.1*F*/(XI_{0})^{0.5}, and the crosslink depth (z*) is approximately 200 to 500 μm, depending on the UV light dose range of 1.0 to 3.0 J/cm^{2}. The measured data of RGX (ex vivo) by Bekesi et al.^{1}with z* = 100 μm are much smaller than UVX. However, the protocols and mechanisms in RGX (dominated by a type II CXL) and in UVX (dominated by a type I CXL^{4}) are different and cannot be fairly compared. Our S-formula shows the CXL strength (or density) in type I is proportional to^{5}*X*= exp(−*A*z). However, both are decreasing function of the light intensity (I_{0}) and increasing function of the diffusion depth (D), suggesting a greater density (strength) of crosslinks produced by RGX than UVX, as indicated clinically by Bekesi et al.^{1}Furthermore, the CXL efficacy may be strongly influenced by the frequency of RF instilled during the UV exposure (defined as Fdrop) to be detailed as follows. The Fdrop in RGX and UVX are different in the study of Bekesi et al.^{1}To overcome the drawback of low efficacy in accelerated CXL as predicted by the S-formula, a RF CCM is proposed as follows. In the conventional Dresden protocol, extra RF drops were instilled during the UV exposure (with a frequency Fdrop = 5–10), which reduced the effective dose from 5.4 J/cm

^{2}to approximately 4.0 J/cm^{2}, based on our calculations.^{2,3}For an optimal protocol (for fast and efficient CXL in the anterior stroma), I propose Fdrop = 1 to 4 to compensate the fast RF depletion in the anterior stroma, especially in high intensity (>18 mW/cm^{2}). In contrast to the conventional Dresden protocol, which keeps the RF in a saturated condition during the UV exposure, CCM proposes to turn off the UV light after each of the extra RF drops applied to the stroma and waiting for a period approximately 1.0 to 2.0 minutes to allow enough RF diffusion (with a diffusion depth D > 150 μm) before it is turned on again. In the above proposed CCM, my theory predicts comparable efficacy (for the same dose) for intensity of 1.5 to 45 mW/cm^{2}, based on a combined efficacy formula defined as c-Ceff = 1− exp [−(S1 + S2 + … Sj)], with j = Fdrop, and Fdrop is given by the integer portion of square root of [I_{0}/3], that is, Fdrop = (1, 1, 2, 3, 3, 4), for I_{0}= (1.5, 3, 9, 18, 30, 45) mW/cm^{2}and exposure time t = (30, 30, 10, 5, 3, 2) minutes. The above CCM proposes that higher intensity requires larger Fdrop (or more RF resupply) to compensate the faster bleaching effect in the anterior stroma (100–250 μm), which is re-treated by Fdrop times, and the waiting period (with UV off) after each RF drops secures enough diffusing depth (D > 150 μm). Numerical simulation of c-Ceff (to be shown elsewhere) under the new CCM protocol shows a stronger correlation with the measured data of Wernli et al.^{3}than the simple protocol (with Fdrop = 0) or Dresden protocol (with Fdrop > 5). The Figure shows an example of c-Ceff for various UV intensities of 3 to 45 mW/cm^{2}, with Fdrop = (1, 2, 3, 3, 4), where all cases have efficacy above the threshold value 80% within the anterior stroma crosslinked depth range of 0 to 200 μm. In comparison, curves in the range 400 to 500 μm are associated with the situation of Fdrop = 0, which shows low efficacy < 80% for high intensity with I_{0}> 18 mW/ cm^{2}, where diffusion depth D = 500, 200, 150, and 150 μm were used for Fdrop = 1, 2, 3, and 4. The cutoff (or maximum) intensity with c-Ceff < 80% (within 0–200 μm stroma) predicted by the S-formula is approximately 45 to 55 mW/cm^{2}, consistent with the clinical data of Wernli et al.^{4}The above formulas also demonstrate that not only crosslink depth (z* > 150 μm) but also crosslink strength (S* > 1.6, or c-Ceff > 0.8) is required in order to achieve high crosslinked stroma volume which is proportional to (z*S*), as also suggested by Bekesi et al.^{1}To conclude, the theoretically proposed CCM using an accelerated CXL while keeping efficacy similar to that of the conventional CXL requires much more basic clinical study to validate the multiple factors influencing the CXL efficacy, including C

_{0}, F, I_{0}, D, and Fdrop, and the associated cutoff maximum intensity (I*) and minimum exposure time (T*), as shown by the S- and z* formulas for type I CXL. Greater details of photochemical kinetics of type II CXL and its influencing factors of efficacy, such as oxygen environment and the generation of reactive oxygen species (ROS), were shown elsewhere.^{5}Many debating issues about the CXL efficacy, such as pulsing versus continuous wave operation, accelerated versus conventional CXL, the minimum corneal thickness, and the role of oxygen in types I and II CXL. require further clinical studies, although they have been partially resolved theoretically.^{3}References

Bekesi
N,
Gallego-Muñoz
P,
Ibarés-Frías
L,
et al.
Biomechanical changes after in vivo collagen cross-linking with rose bengal–green light and riboflavin-UVA.

*Invest Ophthalmol Vis Sci*. 2017; 58: 1612– 1620.
Lin
JT,
Cheng
DC.
Modeling the efficacy profiles of UV-light activated corneal collagen crosslinking.

*PLoS One*. 2017; 12: e0175002.
Lin
JT
.
A critical review on the kinetics, efficacy, safety, nonlinear law and optimal protocols of corneal cross-linking.

*J Ophthalmol Vis Neurosci*. 2018; 3: 017.
Wernli
J,
Schumacher
S,
Spoer
E,
Mrochen
M.
The efficacy of corneal cross-linking shows a sudden decrease with very high intensity UV light and short treatment time.

*Invest Ophthalmol Vis Sci*. 2013; 54: 1176– 1180.
Lin
JT.
Photochemical kinetic modeling for oxygen-enhanced UV-light-activated corneal collagen crosslinking.

*Ophthalmol Res*. 2017; 7: 1– 8.