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

Aims: To resolve the controversial issues of UV-light-initiated corneal collagen cross-linking (CXL) by theoretical formulas and measured clinical outcomes. Study Design: Analysis and measured data of CXL. Place and Duration of Study: New Vision Inc, Taipei, between June 2021 and August 2021. Methodology: The controversial issues are addressed and resolved by analytical formulas including: the validation of Bunsen Roscoe law (BRL), the cutoff light intensity, the minimum corneal thickness, the demarcation line depth, the role of oxygen and riboflavin concentration. The overall CXL efficacy is governed by UV-A light intensity, dose, exposure time, mode of exposure (pulsed or CW), the riboflavin concentration, diffusion and drops pre-operation and interoperation administration, the concentration of oxygen in the stromal tissue (pre-op and inter-op), and environmental conditions. The length of the riboflavin presoaking time and viscosity of the riboflavin film also affect the crosslink depth. Analytic formulas are derived for the scaling laws for type-I and type-II efficacy, given by the square root of light intensity, and light dose, respectively. Conclusion: The controversial issues of CXL may be partially resolved via analytic formulas, and compared with measurements. The scaling laws of type-I and type-II efficacy are different and given by analytic formulas. Our formulas also predict the maximum light intensity and the minimum corneal thickness, which are consistent with measurements. Original Research Article Lin; OR, 15(1): 23-34, 2021; Article no.OR.73526 24


INTRODUCTION
UVA light (at 365 nm) and riboflavin photosensitized corneal collagen cross-linking (CXL) was first proposed by Spoerl et al. [1,2] in1998 to increase the corneal biomechanical strength and stabilize the ectatic cornea. The standard Dresden (SD) protocol was proposed (in 2003) by Wollensak et al. [3], in which a UVA light intensity of 3.0 mW/cm 2 was applied to the cornea for an irradiation time of 30 minutes, such that a light fluence (dose) of 5.4 J/cm 2 was delivered to the cornea. To shorten the irradiation time of the SD protocol, accelerated CXL was also developed [4,5], based on Bunsen and Roscoe law [6], leading to the AC protocol given by light intensity of I= (3,9,18,30,45) mW/cm 2 , with the associated irradiation time is inverse proportional to the light intensity given by t= (30,10,5,3,2) minutes, such that the total dose applied to the cornea is fixed at 5.4 J/cm 2 [1]. In the past 18 years (2003-2020], 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]: (a) Validation of SD protocol for the minimum (safety) corneal thickness of 400um, and a required light dose of 5.4 J/cm 2 proposed by Spoerl et al. [1][2][3]. The threshold damage dose of the endothelium (0.63 J/cm 2 ) proposed by Wollensak et al [28] was largely underestimated, compared to the more recent data of Mooren et al. [29]; and leading to an overestimated minimum thickness of 400 um; (b) Validation of the Bunsen and Roscoe law (BRL) of reciprocity [27] for accelerated CXL (AC) protocol. The new efficacy scaling law developed by Lin, a nonlinear law replacing the linear law based on BRL; (c) For a fixed light dose, AC was clinically reported to be less efficient than SD [4,5], and having a maximum light intensity limitation [30]; (d) The role of oxygen in type-I and type-II CXL [31][32][33][34]; (e) Strategies for improved CXL efficacy and optimal protocols.
The present article intends to resolve the abovelisted issues totally (if theoretically), and partially (if by experimental evidence). Comprehensive and updated formulas will be used to analyze, support, or predict the recently reported clinical outcomes including that of Hafez et al. [43] regarding minimum cornea thickness, DLD; the role of riboflavin (RF) concentration reported by O'Brart et al. [41]. The formula for cut-off light intensity is derived and compared with the reported data by Wernli et al. [30]. New scaling laws for CXL efficacy for both type-I and type-II will be developed. Analytic formulas for CXL efficacy is updated for the influencing factors of oxygen, riboflavin concentration, light intensity, light dose, corneal thickness, and kinetic rate constants.

Crosslink Rate Equation
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]. We note that Eq. (1) is much more complete than previous models of Kamave et al. [24], which is the special case when C(t) is a constant using a continuing resupply of RF, or for the case of a perfect regeneration (REG) of RF. They also ignored the type-I major term of K 1 R. Schumacher et al 25] and Semchishen et al. [26] are limited to type-I conversion, K 1 R, and ignored k 3 T term and the oxygen-mediated term, K 2 S in our Eq. (1). They also ignored the RGE cycle effects. The modeling of Kling et al. [27] is based on Kamave [24], but only showed the algorithm for numerical calculations without analytic formulas. Comparing to the above described previous modeling [24][25][26][27], our modeling, shown by Eq. (1), is the most complete and accurate one.
The quasi-steady-state solutions are given by [21,22]

The Efficacy Formulas
Considering a non-perfect REG case with C(t)=C 0 exp(-d't), with d'=bIg", and g" defining the degree of regeneration (with g"=0, for the perfect case). the approximated analytic solution of Eq.
(3) leads to where k=bIC 0 /A 0 is the contribution from the K 3 T term; , and H'(t)=1/d", which is proportional to [C 0 /(bI)] 0.5 , an increasing function of C 0 , but decreasing function of the light intensity, I(z)=I 0 exp(-A'z). Therefore, higher light intensity leads to lower steady state conversion than that of lower light intensity. This feature will be shown later in Fig.3, in comparing to type-II. We note that the OIH effect plays no role in this case (ii) of type-I process. This steady state feature may be also used to analyze the maximum light intensity feature clinically measured by of Wernli et al. [14], where a sudden efficacy decrease at high light intensity (about 65 mW/cm 2 ).
For type-II dominant case, we need to solve for oxygen concentration, X(t), given by the kinetic equation [22] = −(k"R + k T)X + k S + P 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] Time integral of Eq. (1) (for K 1 =K 3 =0), only the K 2 S term, with S=(k 4 /k 6 )g'TX(t) and X(t) given by Eq. (6), we obtain where p'=(k 4 K 2 /(k 6 K 3 )(g'T'X 0 ), and T'=bIC 0 . Above CE has a transient state CE=p't, which is proportional to the light dose (I 0 t), but a steadystate CE=p'/D, which is almost independent of light intensity. Our Eq. (6) and (7) may be compared with the Eq. (5) and (11) of Kling et al [5], however their formulas are not expressive forms, including unknown parameter [EM] in the equations and can be solved only numerically. They also ignored the type-I process, and assumed the oxygen-mediated type-II is dominant.

The Maximum Light Intensity
The maximal (or cutoff) light intensity reported by Wernli et al. [30] may be analyzed by the following. We will use type-I efficacy, Eq, (4), let the steady-state CE equals a minimum CE=E TH , a steady-state threshold CE for sufficient CXL.
, we obtain the cutoff maximum intensity I 0 * (on surface, z=0) where G'=2K 1 (bC 0 /k') 0.5 /(bg"), and E"=1/(1-E TH ). The above formula defines the cutoff light intensity proportional to the ratio of C 0 and lnE'. For a given C 0 =0.1 to 0.4%, the predicted range of I 0 * is about 45 to 65 mW/cm 2 , to be shown later.

Minimum Thickness
To estimate the safe dose and minimum thickness, we need to find the time (t) and depth (z) dependence RF concentration, C(t,z), and light intensity, I(z,t). Lin [11] has developed a time-dependent Beer-Lambert law (BLL) for the light intensity given by I(z,t)= I 0 exp[-(q'-Bt)z], with q'=2.3(a'C 0 +Q) is an effective absorption constant and B=2.3C 0 (a'-b')d', and an approximated C(z,t)=C 0 exp(-d't), with d'=bIg"t. Where a'=204 (1/%/cm) and b' (an unknown value) are the extinction coefficients of RF and the photolysis product, respectively; Q=13.9 (1/cm) is the absorption coefficient of the stroma.

Demarcation Line Depth (DLD)
DLD may be defined by when CE is larger than an efficacy threshold value (E') for collagen tissue to be effectively affected to form the DLD. For type-I of Eq. (4), the transient state, , and ignored kH'<<1, and let CE=E', or H=ln[1/(1-E')], we obtain (for type-I) , and E"=1/(1-E'), which has a similar tendency as that of Z*, both are increasing function of ln(E 0 ), but a decreasing function of the damage threshold dose (E'). However, Z' is also inversely proportional to the square root of light intensity (for a given light dose). That is higher intensity has a smaller DLD, consistent with the measured data [44]. We may rewrite Eq. for I 0 in mW/cm 2 . and Z' in um.
As expected, Eq. (10) and (11) for the Z' (or DLD) have a similar trend as that of Z*, Eq. (10), both are increasing functions of ln(E 0 ). We may rewrite Z'=(1/q')lnR', with R'= KE 0 /lnE", with K=K' for type-I, and K=K" for type-II. The actual value of Z' may be calculated if the E' value can be measured accurately at a reference point.

Scaling Laws
From our theoretical formulas, we summarize the following important scaling laws, which give us the role of each of the influencing factors and their nonlinear relationships.
(i) Efficacy of type-I CXL given by Eq. (4) In comparison, steady state CE=1 which has a scaling power of intensity, same as Eq. (12.a), but independent to E 0 (or dose).
(ii) For type-II CXL, given by Eq. (10), has a transient state of CE=1-exp(-X't), with X'= m'[C 0 I 0 X 0 ], having a linear scaling law of X't=C 0 E 0 X 0 , which is proportional to the light dose and oxygen initial concentration (X 0 ). In comparison, steady-state CE=1-exp(-KX 0 ), which is almost independent of the light dose, or RF concentration and it is solely depending on which has a scaling power of -0.5 for light but independent to II CXL, given by Eq. (10), has a X't), with X'= ], having a linear scaling law of (13.a) to the light dose and ). In comparison, ), which is almost independent of the light dose, or RF concentration and it is solely depending on oxygen initial concentration (X 0 ), given by a scaling law of CE=KX 0 (1-0.5 KX 0 + KX 0 /6 +...) (iii) Scaling law for minimum corneal thickness, from Eq. (9), Z*=(1/q') ln(R), with R= E 0 /E* (iv) Scaling law for DLD is given by, from Eq. and from Eq. (11) for type-II, We note that both Z* and Z' have similar functional forms proportional to ln (R), with R proportional to the light dose (E 0 ).

Analysis of Maximum Light Intensity
A maximum (cutoff) light intensity (MLI) was reported by Wernli et al. [30] that the efficacy of CXL shows a sudden decrease with very high intensity UV light. This feature may be analyzed by our theoretical curve in Fig. 1 (iv) Scaling law for DLD is given by, from Eq. We note that both Z* and Z' have similar functional forms proportional to ln (R), with R

Maximum Light Intensity
A maximum (cutoff) light intensity (MLI) was [30] that the efficacy of CXL shows a sudden decrease with very highintensity UV light. This feature may be analyzed Fig. 1

Analysis of Thin Corneas
The recent clinical works of Hafez et al showed safety cases for very thin corneas of (214 to 390 um), much less than the conventional minimum criterion of 400 um (based on the Dresden protocol). Our formula demonstrates the theoretical minimum thickness (Z*) could be as thin as 100 um (after epithelium removed), as far as the applied dose (E than the threshold value (E*  [43] at z=214 um, corresponding to our R=8, or a dose of E 0 =8E*; and z=400 um, for R=45. However, the actual value of E* and the referenced rat require further clinical measurements. [43] reported the non-linear relation between the UV irradiation time and predicted demarcation line (referred to their Fig. 3). Fig. 2 shows calculated curves based on Z*= (1/q')ln(R), with q'=2.3(a'C 0 +Q) is a decreasing function of RF concentration, C 0 . That is, a higher concentration has better safety protection such that thinner cornea is allowed. In contrast, higher dose ( larger R=E 0 /E*) requires a thicker cornea. For example, for Z*= (16.5/C 0 )ln(R), for R=12, Z*=(100,200,400) um, for C 0 =(0.1, 0.2, 0.4)%. 28 recent clinical works of Hafez et al. [43] showed safety cases for very thin corneas of (214 to 390 um), much less than the conventional minimum criterion of 400 um (based on the Dresden protocol). Our formula demonstrates the theoretical minimum thickness (Z*) could be as thin as 100 um (after epithelium removed), as far as the applied dose (E 0 ) is less than the threshold value (E*). For example, for the case of Bzt=0, Eq. (9) becomes Z*=(1/q') /E*. For example, for 1/%/cm), Q=13.9 (1/cm), we +32=79(1/cm)=0.0079(1/um), =0.1%, and Z*=127ln(R  [43] at z=214 um, corresponding to our R=8, or a dose of =8E*; and z=400 um, for R=45. However, the actual value of E* and the referenced ratio (R') require further clinical measurements. Hafez et al.

Fig. 2. Minimum corneal thickness (Z*) versus threshold for RF concentration, C
linear relation between the UV irradiation time and predicted demarcation 3). Fig. 2 shows calculated curves based on Z*= (1/q')ln(R), with +Q) is a decreasing function of RF . That is, a higher concentration has better safety protection such that thinner cornea is allowed. In contrast, higher dose (or /E*) requires a thicker cornea. For )ln(R), for R=12, =(0.1, 0.2, 0.4)%.
Our Z* formula is more accurate and expressive (in analytic form) than that of Kling et al which showed a complex algorithm requiring numerical solution, and was limited to the type process. also showed the Z* vs. UV irradiation time (t), or dose for fixed light intensity. However, their data were based on the Dresden protocol of t=30 minutes, for 3 mW/cm 2 intensity, and estimated damage dose threshold (E*). Therefore, our formula based on the relative ratio of E 0 /E* is much more accurate (if E* could be measured accurately). Fig. 3 of Hafez et al. [43] may be compared with our formula, Eq. (10), Z*=(1/q')ln(KE 0 /lnE'), which is proportional to ln(irradiation time), for fixed light intensity (I and it is an increasing function of ln(E a consistent trend as their Fig. 3.

Minimum corneal thickness (Z*) versus threshold ratio R=E 0 /E*, based on Z*= (1/q')ln(R),
for RF concentration, C 0 = (0.1, 0.2, 0.4)%, for curves (1,2,3) ; Article no. OR.73526 Our Z* formula is more accurate and expressive (in analytic form) than that of Kling et al. [27] ex algorithm requiring numerical solution, and was limited to the type-II  Fig. 4 shows DLD versus G, Z'=(130,200,300) um for a given G=15 and various intensity I= (10,30,60) . Both Figures show that DLD is decreasing function of light intensity, but an increasing function of G factor, noting that G is . Fig. 5 compares d data of Hafez et al. [43], and calculated values based Eq. (10.b). They also showed the Z* vs. UV irradiation time (t), or dose for fixed light intensity. However, their data were based on the Dresden protocol of t=30 intensity, and an underestimated damage dose threshold (E*). Therefore, our formula based on the relative ratio /E* is much more accurate (if E* could be measured accurately). Fig. 3 of Hafez et al. [43] may be compared with our formula, Eq.
30 [41] reported that efficacy is an increasing function of RF concentration (in the Clinical studies of Lang et al. [5] showed less efficacy than standard low ) CXL for the same fluence (dose) based on BRL. To overcome this intrinsic [16,20] recently proposed a new protocol called the riboflavin concentration controlled method (CCM) to improve the efficacy of AC by supplemental RF during the UV exposure to compensate for fast depletion of RF by UV light. However, RF concentration, C(t), compensated due to the regeneration effect (RGE), in which C(t) = , for a perfect RGE case with g"=0, which also depends on the type of photoinitiator. The optimal concentration was also theoretically proposed, specially for the efficacy in the stroma (with z>0) [11]. From Eq. state efficacy given by H = , which is an increasing function . To show this scaling law, Fig. 6 shows the calculated efficacy curves based on a exp(-m[C 0 /I 0 ] 0.5 ), with m=10 as a fit parameter comparing to measured data. For a fixed light intensity, efficacy is an increasing function of C 0 0.5 , but a . For examples, (shown by the red lots of Fig.6), efficacy CE=(0.8, 0.7, 0.6), for I= (9,18,30) mW/cm 2 , at a given ured data of O'Brart et al. [41] are fit to our calculated curve. We note that: As discussed in our previous paper [22], constant if there is a continuing resupply of RF, or for the case of a perfect REG of RF (or a catalytic cycle). To demonstrate the RF depletion, we conducted a measurement of RF (mixed with pure water) under UV irradiation at various RF concentrations, in which the increase of light intensity passing through the RF solution (path of 10 mm) was recorded indicati decrease of RF which also has color changes. This feature can be explored by I(z,t)=I q'+Bt]z, a revised time-dependent BLL [11,17 with B=2.3(a'-b')C 0 d', proportional to the depletion rate of RF (d') and its initial concentration C 0 . If d'=0, then I(t,z) is time independent. As shown by Fig. 7, high RF concentration has a larger light intensity increasing slope, defined by B. As discussed in our previous paper [22], C(t) is a constant if there is a continuing resupply of RF, or for the case of a perfect REG of RF (or a ate the RF depletion, we conducted a measurement of RF (mixed with pure water) under UV irradiation at various RF concentrations, in which the increase of light intensity passing through the RF solution (path of 10 mm) was recorded indicating the decrease of RF which also has color changes. This feature can be explored by I(z,t)=I 0 exp[dependent BLL [11,17], d', proportional to the depletion rate of RF (d') and its initial then I(t,z) is timeindependent. As shown by Fig. 7, high RF concentration has a larger light intensity

Summary of up-dated CXL Features
Factors influencing the CXL efficacy include: UV A light intensity, dose, exposure time, mode of exposure (pulsed or CW), riboflavin concentration, diffusion and drops pre and interoperation administration, the concentration of oxygen in the stromal tissue (pre-op and inter-op), and environmental conditions. The length of the riboflavin presoaking time and viscosity of the ri film also affect the crosslink depth. From our analytic formulas, Eq. (4) to (11), and scaling laws, Eq. (11) to (13), the key features of type and type-II CXL are summarized and compared as follows:

dated CXL Features
Factors influencing the CXL efficacy include: UV-A light intensity, dose, exposure time, mode of exposure (pulsed or CW), riboflavin concentration, diffusion and drops pre-operation and interoperation administration, the concentration of oxygen in the stromal tissue op), and environmental conditions. The length of the riboflavin presoaking time and viscosity of the riboflavin film also affect the crosslink depth. From our analytic formulas, Eq. (4) to (11), and scaling laws, Eq. (11) to (13), the key features of type-I II CXL are summarized and compared (a) Type-I and type-II coexist in CXL, in the presence of oxygen. However, there is no type-II when oxygen is depleted or in a condition without oxygen. (b) Type-I has two cases: case (i) with unimolecular termination, the radical (R) and efficacy are a linear increasing function of bIgC/G, but they are decr function of oxygen due to the OIH effect which reduces the radical (R) and the efficacy, because G is an increasing function of oxygen (or X),

Light intensity versus irradiation time (t) based on nonlinear law (shown in red dots),
II coexist in CXL, in the sence of oxygen. However, there is no II when oxygen is depleted or in a case (i) with unimolecular termination, the radical (R) and efficacy are a linear increasing function of bIgC/G, but they are decreasing function of oxygen due to the OIH effect which reduces the radical (R) and the efficacy, because G is an increasing function of oxygen (or X), G= G= k"X+ [A], as shown in Eq. (2). In comparison, ase (ii) for bimolecular termination, R is root function of , but OIH plays no role. for steady-state. (e) The scaling laws of type-II efficacy given by, Eq. (13): X't=C 0 E 0 X 0 for transient state, and (f) steady-state CE=1-exp(-KX 0 ), which is almost independent to the light dose, or RF concentration and it is solely depending on oxygen initial concentration (X 0 ), given by a scaling law of CE=KX 0 (1-0.5 KX 0 + KX 0 /6 +...). (g) Oxygen is required for oxygen-mediated (OM) type-II but it is not required in type-I. Therefore, type-II only plays a limited and transient state role for t<t 0 , with t 0 being the depletion time of oxygen. In the transient stage (about 3 to 20 seconds), both type-I and type-II coexist until the oxygen is depleted; then type-I dominates before the oxygen is resupplied or replenished. It was reported by Kamaev et al. [24] that it took about 3 to 4 minutes for oxygen replenishment under a normal environment. Therefore, the improved CE by a fast pulsed mode is unlikely to occur [35][36][37][38], unless the light on-off time period is few minutes. (h) RF depletion in type-I is compensated by the REG term which is a decreasing function of oxygen. For the case of perfect REG, RF is a constant due to the catalytic cycle. (i) Wernli et al. [14] reported a sudden efficacy decrease at high light intensity (about 65 mW/cm 2 ), is also predicted by our formula, Eq. (8), and Fig 1, that the steady-state efficacy is a decreasing function of light intensity, and a sudden drop is expected when the efficacy is below an efficacy threshold. (j) The minimum corneal thickness (Z*) and the DLD (Z'), are given by formula Z*=(1'q') ln(R'), with R'=KE 0 /E*; and Z'=(1/q') ln(R'), with R'= KE 0 /lnE", respectively, where both are increasing function of ln(E 0 ). (k) For high CE and typical DLD of 200 to 250 um, sufficient pre-operation and interoperation administration of RF are required, such that the RF initial diffusion depth is at least 200 um in the stroma and the RF depletion could be compensated by our proposed CCM.

CONCLUSION
The controversial issues of CXL are partially resolved via our analytic formulas, and compared with measurements. The roles of RF concentration and oxygen in the CXL efficacy could be analytically expressed. The scaling laws of type-I and type-II efficacy are different and given by analytic formulas. Our formulas also predict the maximum light intensity and the minimum corneal thickness, which are consistent with measurements.

CONSENT AND ETHICAL APPROVAL
It is not applicable.