Critical Analysis of Corneal Cross-linking (Part-I): Formulas for Efficacy, Safety Dose, Minimum Thickness, Demarcation Line Depth and the Role of Oxygen

Abstract

Purpose: To update and derive formulas for the efficacy and kinetics of corneal collagen crosslinking (CXL) including both type-I and oxygen-mediated type-II mechanisms, the role of oxygen, the initiator regeneration, safety dose, minimum corneal thickness and demarcation line depth.
 Study Design: Modeling the kinetics of CXL in UV light and using riboflavin as the photosensitizer.
 Place and Duration of Study: Taipei, Taiwan, between June, 2021 and July, 2021.
 Methodology: Coupled kinetic equations are derived under the quasi-steady state condition for the 2-pathway mechanisms of CXL. For type-I CXL, the riboflavin (RF) triplet state [T] may interact directly with the stroma collagen substrate [A] to form radical (R) and regenerate initiator. For type-II process, [T] interacts with oxygen to form a singlet oxygen [1O2]. Both reactive radical (R) and [1O2], can interact with the substrate [A]) for crosslinking. Based on a safety dose and a threshold dose, formulas for the minimum corneal thickness and demarcation line depth (DLD) are derived.
 Results: Our updated theory/modeling showed that oxygen plays a limited and transient role in the process, in consistent with that of Kamave. In contrary, Kling et al believed that type-II is the predominant mechanism, which however conflicting with the epi-on CXL results. For both type-I and type-II, a transient state conversion (crosslink) efficacy in an increasing function of light intensity (or dose), whereas, its steady state efficacy is a deceasing function of light intensity. RF depletion in type-I is compensated by the RF regeneration term (RGE) which is a decreasing function of oxygen. For the case of perfect regeneration case (or when oxygen=0), RF is a constant due to the catalytic cycle. Unlike the conventional Dresden rule of 400 um thickness, thin cornea CXL is still safe as far as the dose is under a threshold dose (E*), based on our minimum thickness formula (Z*). Our formula for thin cornea is also clinically shown by Hafez et al for ultra thin (214 nm) CXL.
 Conclusion: For both type-I and type-II, the transient state conversion (crosslink) efficacy in an increasing function of light intensity (or dose), whereas, its steady state efficacy is a deceasing function of light intensity. CXL for ultra thin corneas are still safe, as far as it is under a threshold dose (E*), based on our minimum thickness (Z*) formula, which has a similar tend as that of demarcation line depth (Z').

Critical Analysis of Corneal Cross-linking (Part-II): Resolving the Controversial Issues (Theory versus Measurements)

“…Using the shorthand notations: C and T for the RF ground and excited triplet state; R for the active radical, S for the singlet oxygen [ [22,23].…”

“…For the case that there is no external resupplied oxygen (or P=0), and using the first-order solution with [A]=A 0 in the function of g and S=k 4 g'TX, with T=T'/(K 3 A 0 ), and a timeindependent g'=1/ (k 6 + k 1 C 0 + K 2 A 0 ), we obtain [22,23]…”

“…In the past 18 years (2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017)(2018)(2019)(2020), over 1,000 articles were published on the clinical aspects of CXL. In comparison, only about 20 articles were published regarding the basic kinetics and/or the theory of CXL , in which over 75% was published by Lin et al [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. With the lack of indepth fundamental studies and lack of conclusive theoretical predictions, the following controversial (clinical/theoretical) issues remained to be resolved [15,22]:…”

Critical Analysis of Corneal Cross-Linking (Part-II): Resolving the Controversial Issues (Theory Versus Measurements)<strong> </strong>

“…In the past 18 years (2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017)(2018)(2019)(2020), over 1,000 articles were published on the clinical aspects of CXL. In comparison, only about 20 articles were published regarding to the basic kinetics and/or the theory of CXL , in which over 75% was published by Lin et al [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. The lack of in-depth fundamental studies and lack of conclusive theoretical predictions, the following controversial (clinical/theoretical) issues remained to be resolved [15,22]:…”

“…(h) Validation of SD based on a dose of 5.4 J/cm 2 , comparing to the recent clinical data of King et al [42] using a lower dose (2.7 J/cm 2 ), which is half of SD. (i) New criteria for minimum corneal thickness (Z*), in which sub400um thin corneas (214 -398 um) was reported by Hafez et al [43]; (j) Factors influencing the demarcation line depth (DLD) (k) Comparing the Theory/formulas of Z* and DLD, developed by Lin [12,23], Kling et al [27] and Mazzotta et al [44]. (l) The new efficacy scaling law developed by Lin, a nonlinear law replacing the linear law based on BRL; (m) Strategies for improved CXL efficacy and optimal protocols.…”