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Modeling a new strategy and influencing factors for improved efficacy of accelerated corneal crosslinking

Lin, Jui-Teng PhD

Journal of Cataract & Refractive Surgery: April 2019 - Volume 45 - Issue 4 - p 527-529
doi: 10.1016/j.jcrs.2019.02.001
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Clinical studies by Ng et al.1 and the theoretic formulas of Lin and Cheng,2 and Lin3 showed that accelerated corneal crosslinking (CXL) is less efficacious than CXL using the standard Dresden protocol for the same fluence (dose). To improve the efficacy of accelerated CXL, Lin3 proposed a new protocol in which supplemental riboflavin is used during ultraviolet-A (UVA) light exposure to compensate for the fast depletion of riboflavin by UVA light and a higher riboflavin concentration is used.4,5 The conventional Dresden safety criteria (5.4 mJ/cm2 and minimum corneal thickness 400 μm) was challenged by a new theoretic safety criterion proposed by Lin6 and a recent animal study by Kling and Hafezi,7 in which the UVA fluence was reduced from 5.4 mJ/cm2 to 2.7 mJ/cm2. A second challenge regarding the validity of the Bunsen-Roscoe reciprocal law was reported clinically by Wernli et al.8 and theoretically by Lin,2,9 who reported a maximum UVA light intensity and minimum exposure time. Lin9 recently analyzed the relationship between the efficacy of accelerated CXL and the demarcation line depth; he found that the demarcation line depth dropped suddenly at high intensity. However, the optimum CXL parameters and associated protocols remain controversial.

This paper intends to build a bridge, for the first time, between clinical data and theoretic formulas based on the key factors of CXL. An extended fluence and higher riboflavin concentration for improved accelerated CXL efficacy will be analyzed and compared with the clinically reported results of O’Brart et al.4 and Wernli et al.8

Methods and discussion

Key Parameters

Figure 1 shows formulas and the abbreviations of the key parameters discussed in this paper.2,9,10

Figure 1
Figure 1:
Formulas and abbreviations of key parameters (CCM = concentration-controlled method; CXL = corneal crosslinking).2,9,10

Role of Riboflavin Concentration

In a proposed riboflavin concentration–controlled method by Lin,3 the crosslink time is T* = (4/b) exp(Az), where b = 0.31k (k = 0.01 and is an empirically fit estimated rate constant); A is numerically fit effective absorption constant3 (A = 0.5 x 2.3 m(a’+b’) C0 + Q, where m is a fit constant). During UVA exposure, resupply of riboflavin at every crosslink time (T*) having a frequency is given by3 Fdrop = N − 1, with N = 0.365 [I0/C0]0.5 and N = 2 was chosen, for I0 = 3 mW/cm2 and C0 = 0.1%, as the reference. The riboflavin concentration–controlled method proposes Fdrop = (0,1,2,3,4,5), for a light intensity of I0 = (1.5, 3.0, 9.0, 18, 30, 45) mW/cm2.

The extended fluence (dose) for accelerated CXL (E′) is defined by extending the exposure time of the Bunsen-Roscoe reciprocal law to achieve the same efficacy as the referenced dose (E30, for I0 = 3 mW/cm2), given by E′ = RE30, with the dose ratio R = 0.183 exp(Az)(I0/C0)0.5 based on Lin’s S-formula.3,4 As shown in Figure 2, R = 1.0 at the referenced point (at z = 0), with I0 = 3 mW/cm2 and C0 = 0.1%; R decreases to 0.5, fiving a higher C0 (0.4%). For a fixed C0, R is an increasing function of the light intensity and corneal thickness (z). A higher riboflavin concentration (for z = 0, with a smaller R) also offers higher CXL efficacy, as clinically4 and theoretically5 reported. However, for z > 0, the optimum riboflavin concentration is approximately C0 = 0.28%, which gives a minimum R, or an extended dose.

Figure 2
Figure 2:
Extended dose ratio R versus riboflavin concentration (C) at z = 0 (left) and z = 125 μm (right) for intensity I0 = (3, 9, 18, 30) mW/cm2 (for curves 1, 2, 3,4).

The optimum riboflavin concentration (C*) can also be found by the maximum of the transient S-function,2 or by dS/dC0 = 0, leading to C* = (140mz)−1, with S (for small dose) being proportional to [(C0I0)exp(−Az)]0.5, which defines the CXL efficacy,2 Eff = 1 − exp(−S).

Maximum Light Intensity

A reported limitation of the Bunsen-Roscoe reciprocal law regards the nonlinear feature of CXL efficacy2,8 and the maximum UVA intensity (I*), which can be defined in various ways. When the threshold of the steady state S-function2 (at z = 0) is larger than a threshold value given by d = [Sd/(0.87S0)]2, which leads to I* = C0/d. Therefore, I* = (40, 60) mW/cm2, for C0 = (0.2, 0.3) % and d=0.005. For a fixed C0 = 0.3%, I* = (50, 60, 75) mW/cm2, for d = (0.006, 0.005, 0.004). We may also define I* by the Rf resupply frequency Fdrop = (N-1), with N given by3 N=0.365(I*/C0)0.5. Therefore, I* = 7.5N2C0. For a maximum N=5.0, we obtain I* = (46, 56, 65) mW/cm2, for C0 = (0.25, 0.3, 0.35) %; and for a fixed C0 = 0.3%, we obtain I* = (36, 56, 81) mW/cm2, for N = (4,5,6).

A third method for the maximum intensity (I*) is based on the extended fluence (dose) ratio R = E′/E30, given by3,6 R = 0.182 exp(Az)(I0/C0)0.5, which gives I* = 30.2C0R2. Therefore, I* = (36, 56, 67) mW/cm2 for a maximum R = (2.0, 2.5, 3.0). The R-formula also provides the extended dose having a range of E′ = RE30 = 5.4 to 8.1 J/cm2 if E30 = 2.7 J/cm2 is chosen as the reference dose, as reported by Kling and Hafezi.7 The calculated E′ covers the reported clinical data for improved accelerated CXL,1 with an extended 133% of 5.4 J/cm2, or 7.2 J/cm2, which was also used in the modern protocol of Avedro, Inc. The minimum accelerated CXL exposure time, defined by t* = E′/I*, has a range of 1.80 to 3.75 minutes. The above theoretic values cover the animal data reported by Wernli et al.8; that is, I* = 45 mW/cm2 and t* is approximately 2 minutes.

Crosslink Depth

As presented by Lin,2,5 the crosslink depth, z*, is defined by when the S-function reaches 1−e−M (or 0.87, when M = 2) of its steady-state value and is related to the light dose (E0) by the following: z* = (10000/A) ln [(aqk)E0/M], with z* in μm, E0 in J/cm2. Therefore, extension of the exposure time (t) or light dose (E0 = I0t) for a given light intensity (I0) will increase the crosslink depth, which shows a decrease in the function of the riboflavin concentration resulting from the increase in the absorption constant A = 0.5 x 2.3 m(a’+b’) C0 + Q. A higher riboflavin concentration results in an increased, but more superficial (or small z*) crosslinking effect, as clinically reported by O’Brart et al.4 The efficacy of the extended-dose method is limited by the available riboflavin in the stroma and reaches its steady state when the riboflavin is completely depleted. Extending the exposure time (or light dose) will always achieve a deeper crosslink depth but will not improve the efficacy when it reaches its steady state. Therefore, the extended-dose method is not as efficient as the concentration-controlled method.3

A new concept of CXL efficacy (proposed by Lin5,9) is based on volume efficacy (V), defined by volume = [strength] × [depth], or V = z*S, which requires both z* > 250 um (the efficient anterior stroma), and S > S*, a threshold for efficacy, Eff = 1 − exp(−S) > 0.86, or S > S* = 2.0. The optimum range of riboflavin concentration is governed by the following 2 criteria: (1) minimum crosslink depth (with z* > 250 um), or a maximum C0 (or C2), noting that z* is inversely proportional to A; (2) minimum S-function (with S > S*), or a minimum C0 (or C1), noting that S is proportional to (for the steady state) [(C0/I0) exp(Az)]0.5. The combined conditions lead to the range of riboflavin concentration for efficient CXL as follows: C1 < C0 < C2. More details on the analytic formulas of C1 and C2, and the optimal conditions for maximum efficacy may be found elsewhere.9

Combined Efficacy

The threshold condition of S* > 2.0 might not be achieved, especially in AXL with light intensity > 9.0 mW/cm2, as shown by Figure 3, which has a lower steady-state-efficacy (SSE) than the standard intensity of 3.0 mW/cm2, in which there is no resupply of Rf during the UVA exposure (or Fdrop = 0). The SSE is defined by when the transient term of the S-function reaches its maximum value, ie, E’ = 1.0, as shown by Figure 3. This drawback might be overcome by Lin’s concentration-controlled method, which offers a combined steady-state efficacy given by c − Eff = 1 − exp [−(S1 + S2 + S3+…)], with (S1 + S2 + S3+…) > S*, although each Sj < S*. For a given riboflavin concentration (C0), deeper CXL might be achieved by larger fluence (E0). However, to achieve clinically acceptable CXL efficacy by a minimal E0, one requires an optimum range of C0. For example, C0 = 0.2% to 0.35% and E0 = 2.5 to 3.5 J/cm2 such that [depth], z* = 200 to 300 μm, and [strength], S1 = 1.5 to 2.0, or CXL efficacy Eff = 1 − exp(−S1) = 0.78 to 0.86.

Figure 3
Figure 3:
S-function profiles10 for corneal crosslinking efficacy (at z = 0), intensity I0 = (3, 9, 18, 30) mW/cm2 (curves 1, 2, 3, 4), and riboflavin concentration C0 = 0.1% (t = exposure time).

Recent clinical studies11,12 reported a much lower intrastromal riboflavin concentration (0.01% to 0.03%) than the applied surface riboflavin concentration (0.1% to 0.25%). The above discussed formulas require a revised effective absorption factor given by2 A(z,t)= 2.3 [(204 − b′) C(z,t) + 32], with b′ being the absorption of the photolysis; b′= 50 to 100 (1/cm/%), has a lower value range of 35 to 45 (cm−1) for low C0 = 0.01% compared with the range of 50 to 75 (cm−1) for high C0 = 0.1%. Furthermore, a much lower dose (2.7 to 3.6 J/cm2) than used in the Dresden protocol (5.4 J/cm2) for efficient CXL using lower intensity of 1.5 to 4.0 mW J/cm2 was reported.7,12 Therefore, the extended dose presented in this study may be proportionally reduced. The estimated rate constant (k), defining the crosslink depth (z*) and time (T*) in this study, was empirically fit based on the measured riboflavin concentration profile of Lombardo et al.13 More details about a low dose and low intrastromal riboflavin concentration will be given in a subsequent paper (unpublished).


To conclude, Lin’s formulas, summarized in Figure 1, build a bridge between clinical data and the theoretic ranges of extended fluence for improved accelerated CXL efficacy, maximum intensity, minimum exposure time, and an optimum riboflavin concentration approximately 0.28%. Lin’s riboflavin concentration–controlled method offers a new strategy to achieve high a crosslinked volume. In addition, Lin’s method is more efficient than the extended-dose method, which is limited by the low steady-state efficacy. More accurate predictions would require further clinical studies to confirm certain parameters used in the formulas presented here.


1. Ng ALK, Chan TCY, Cheng ACK. Conventional versus accelerated corneal collagen cross-linking in the treatment of keratoconus. Clin Exp Ophthalmol. 2016;44:8-14.
2. Lin J-T, Cheng D-C. Modeling the efficacy profiles of UV-light activated corneal collagen crosslinking. PLoS ONE. 2017;12(4):e0175002.
3. Lin J-T., 2018. A proposed concentration-controlled new protocol for optimal corneal crosslinking efficacy in the anterior stroma [letter], Invest Ophthalmol Vis Sci, 59, 431-432.
4. O’Brart NAL, O’Brart DPS, Aldahlawi NH, Hayes S, Meek KM. An investigation of the effects of riboflavin concentration on the efficacy of corneal cross-linking using an enzymatic resistance model in porcine corneas. Invest Ophthalmol Vis Sci. 2018;59:1058-1065.
5. Lin J-T., 2018. The role of riboflavin concentration and oxygen in the efficacy and depth of corneal crosslinking [letter], Invest. Ophthalmol Vis Sci, 59, 4449-4450.
6. Lin J-T. Efficacy and Z* formula for minimum corneal thickness in UV-light crosslinking. [letter] Cornea 2017;36:e30-e31, reply by CC aruso, RL Epstein, C Ostacola, G Barbero, e31–e33.
7. Kling S, Hafezi F. Biomechanical stiffening: slow low-irradiance corneal crosslinking versus the standard Dresden protocol. J Cataract Refract Surg. 2017;43:975-979.
8. Wernli J, Schumacher S, Spoerl 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.
9. Lin J-T, Liu HW, Chen KT, Cheng DC. Modeling the optimal conditions for improved efficacy and crosslink depth of photo-initiated polymerization. Polymers. 2019;11:217.
10. Lin J-T. Efficacy S-formula and kinetics of non-oxygen-mediated (type-I) and oxygen-mediated (type-II) corneal cross-linking. Ophthalmol Res. 2018;8(1), article number OR:39089
11. Lombardo G, Micali NL, Villari V, Serrao S, Lombardo M. All-optical method to assess stromal concentration of riboflavin in conventional and accelerated UV-A irradiation of the human cornea. Invest Opthalmol Vis Sci. 2016;57:476-483.
12. Stulting RD, Trattler WB, Woolfson JM, Rubinfeld RS. Corneal crosslinking without epithelial removal. J Cataract Refract Surg. 2018;44:1363-1370.
13. Lombardo G, Villari V, Micali NL, Leone N, Labate C, De Santo MP, Lombardo M. Non-invasive optical method for real-time assessment of intracorneal riboflavin concentration and efficacy of corneal cross-linking. J Biophotonics. 2018;11(7):e201800028.


Dr. Lin is the chief executive officer of New Vision, Inc.

© 2019 by Lippincott Williams & Wilkins, Inc.