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Research Article

A Critical Review on the Kinetics, Efficacy, Safety, Nonlinear Law and Optimal Protocols of Corneal Crosslinking

Jui-Teng Lin

Correspondence Address :

Jui-Teng Lin
New Vision Inc., 5F, No. 27, Lane 10, Jiuquan St. Da-tung Dist. Taipei
Taiwan
Tel: 886-961-306-877
Email: jtlin55@gmail.com

Received on: December 11, 2017, Accepted on: December 22, 2017, Published on: January 02, 2018

Citation: Jui-Teng Lin (2018). A Critical Review on the Kinetics, Efficacy, Safety, Nonlinear Law and Optimal Protocols of Corneal Crosslinking

Copyright: 2018 Jui-Teng Lin. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract
Corneal collagen cross-linking (CXL) mechanisms including the oxygen-mediated (OM) via singlet oxygen or other reactive oxygen species (ROS), and non-oxygen-mediated (NOM) via direct coupling of the triplet riboflavin to the stroma collagen substrate [A] are included in coupled kinetic equations to describe the three pathways involved in CXL. Analytic formulas for the crosslink time, safety criteria, and overall CXL efficacy are derived. I propose that CXL is governed by both type-I and -II mechanisms, with NOM (in type-I) being the predominant contribution, while oxygen only plays a transient role, in contrary to the conventionally believed OM-predominant mechanism. The debating issues solved/analyzed mathematically in this study include: the validation of Bunsen Roscoe law (BRL), the intensity dynamics and its cutoff maximum, the safety criteria, the demarcation depth, the competing of OM and NOM mechanisms and the role of oxygen. The overall CXL efficacy is governed by the UV light intensity and dose (or fluence), the RF and oxygen initial concentration and their diffusion depths, the triplet-state quantum yield, and the kinetic rate constants. A new protocol using concentration-controlled method (CCM) and a nonlinear-revision of the Dresden dose (based on BRL) can largely improve the efficacy in accelerated CXL, which is less efficient than the Dresden (low intensity) CXL under conventional non-controlled methods.

Keywords: Corneal crosslinking, Corneal keratoconus, Efficacy, Kinetic Modeling, Oxygen, Riboflavin, Ultraviolet light, Photodynamic therapy
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Introduction
Corneal collagen cross-linking (CXL) was first proposed in 1998 by Spoerl et al to increase the corneal biomechanical strength and stabilize the ectatic cornea [1]. The standard Dresden (SD) protocol was proposed by Wollensak, et al. [2] in 2003, where a UVA light (at 365 nm) was used to treat cornea 9 mm zone at an intensity of 3.0 mW/cm2 for 30 minutes, delivering a fluence (dose) of 5.4 J/cm2.
To shorten the CXL treatment duration while maintaining the similar CXL efficacy, various accelerated (AC) protocols to replace the SD protocol have been proposed based on the Bunsen and Roscoe law (BRL) of reciprocity [3] stating that the effect of a photo-biological reaction is proportional only to the total irradiation dose (E=It), or the product of intensity (I) and exposure time (t). To achieve the same efficacy, the required exposure time based on BRL is given by t=E/I, which gives the protocol for AC; for example, t= (30, 10, 5, 3, 2) minutes for I= (3,9,18,30,45) mW/cm2. To improve the CXL efficacy, various techniques have been proposed including: pulsed mode operation of the UV light, extra oxygen supply to the corneal surface; and enhancement of the riboflavin diffusion such as diffusion in the de-epithelialized stroma (standard method); diffusion through the epithelium into the stroma (transepithelial method); or direct introduction of riboflavin into the stroma (pocket technique, ring technique, needle technique); and enrichment of riboflavin in the stroma by iontophoresis. In addition to UVA light activated riboflavin, other photosensitizers using blue light (at 430 nm) and green light (at 532 nm) were also proposed [4,5]. Recent extensive reviews on the biomechanical results of CXL, efficacy comparisons of SD and AC protocols and various applications combing CXL and other refractive surgeries were reported by Sachdev and Sachdev [4] and Andreanos, et al. [5]. These reviews covered a wide range of protocols including AC protocols [6-23], pulsed protocol [24-30], transepithelial (epi-on) and iontophoresis-assisted protocols [31-34], and topography-guided, or customized protocol [35-38].
Majority of studies find accelerated CXL is not as effective as the SD conventional protocol in corneal flattening, but they have the similar safety and efficacy [6-23].
Furthermore, Demarcation line (DL) depth is widely used as an indirect marker of CXL efficacy. The DL depth after AC CXL appears to be shallower than that of conventional SD protocol, thus less corneal volume seems to be cross-linked with the AC protocol [26-30]. The CXL efficacy was theoretically proposed (to be discussed later) and proportional to the cross-linked corneal stroma volume (V), defined by the product of cross-link depth (z) and cross-link strength (S), i.e., V=zS. For the same dose, similar z can be achieved by AC and SD, but the cross-linked-S in AC is always smaller than that of SD, thus AC has smaller cross-linked V and efficacy.
Comparing to the extensive clinical efforts [4,5] with over 1,000 published articles (during 2003 to 2017), the basic kinetics, or theories/fundamentals of CXL, have been limited to about 10 published articles (2012 to 2017) by a small group of researchers, including Kamaev, et al. [39], Schumacher, et al. [40], Semchishen, et al. [41], Caruso, et al. [42], Kling, et al. [43] and Lin et al. [44-52].
The previous modeling [39-43] assuming a constant, nondepleted riboflavin (RF) and/or a flat profile, will cause a lot of errors in calculating the profiles of C (z, t), I (z, t) and thus the predicted CXL efficacy profiles. The results predicted by Schumacher, et al. [40], and Kling [43] maybe correct, but only limited to the type-II CXL with neglected RF depletion, or idea case that the RF has a flat profile, and only when the conventional Beer-Lambert law (BLL) for UV light intensity is valid. The conventional accepted safety criteria for minimum corneal thickness z*=400 um, for a dose of 5.4 J/cm2, proposed by Wollensak, et al. [52,53] was meaningless without specifying the RF concentration and its profile. The endothelium damaged threshold was also underestimated at least a factor of 5 per recent clinical data of Mooren, et al. [54].
According to Kamaev, et al. [9] they believe that CXL in the cornea is initiated due to the direct interaction between the substrate and excited RF triplets, with singlet oxygen playing a limited and transient role in the process. In contrary, Kling, et al. [8,55] and McCall et al [14] believed that type-II is the predominant mechanism. If Kling et al would be correct, then all the published results of epi-on CXL would be impossible, since Kling claimed oxygen is a must-element of CXL efficacy. Therefore, I believe, it is most likely that CXL is governed by both type-I (non-oxygenmediated) and type-II (oxygen-mediated), with an intial-state (the first 5 to 20 seconds) predominnat by type-II.
In view of above discussions, the debating issues to be resolved/analyzed mathematically in this study include: the validation of BRL and BLL, the intensity cutoff maximum of AC, the safety criteria, the demarcation depth, the role of oxygen in both type-I and type-II CXL. This study will review the sofar developed basic kinetics of CXL and summarize the analytic formulas developed by Lin [44-52]. Critical comments on previous modeling of others [39-43] will be presented. This study will also present greater details of the optimal protocol recently proposed by Lin [51].

Methods and Modeling systems

Photochemical kinetics of CXL

As shown in Figure 1, the CXL process has three pathways. The RF excited triplet-state [T] can undergo two kinds of reactions. Ground state oxygen may couple to [T] to form either singlet oxygen [1O2], or other reactive radicals (ROS) [O-]. In type-I pathway, [T] can interact directly with the collagen substrate (A); or with the oxygen to generate a superoxide anion (O-). In type- II pathway, [T] interacts with the ground-state oxygen [3O2] to form an excited-state singlet oxygen [1O2] [48]. The contribution ratio of the two types of oxygen-mediated CXL depends on the properties of the photosensitizer (PS) concentration For example, in rose Bengal, type-II is the predominant process with singlet oxygen contributing about 80%, and other ROS contributing about 20%. In comparison, for riboflavin, they are 49% and 1%, respectively [4].
Both type-I and type-II reactions can occur simultaneously, and the ratio between these processes depends on the type of photosensitizers (PS) used, the concentrations of PS, substrate and oxygen, the kinetic rates involved in the process. These factors also influence the overall CXL efficacy, particularly the PS triplet state quantum yield (q). Furthermore, the specific protocols and the methods of RF instillations prior to and during the CXL also affect the short and long term outcomes. The overall CXL efficacy basically is proportional to the time integration of the UV light intensity, I (z, t) and the PS and oxygen concentration, C (z, t), and [3O2]. The efficacy reaches a saturated (steady) state when C (z, t) or [3O2] is depleted by the UV light, where higher intensity depletes C (z, t) and [3O2] faster and therefore reaches a lower steady-state efficacy. The Dresden (at 3 mW/cm2) protocol, therefore, is always more effective than the accelerated CXL with 9 to 45 mW/cm2. However, this drawback maybe overcame by a concentration-controlled method (CCM) first proposed by Lin [51].

Modeling type-I system

We will first present a comprehensive theory to analyze most of the CXL basic features, and more complex theory will be shown later. The kinetic of CXL is governed by both oxygen-mediated (OM) and non-oxygen-mediated (NOM) reactions, in which there are 3 pathways for crosslinking (greater details will be shown later). We will first discuss type-I CXL which is less complex than type-II.
For a CXL modeling system, a UV light propagating in the corneal/stroma depth direction (z), having an initial riboflavin (RF) concentration of C (z, t=0) =C0F(z), the coupled dynamic equation (for type-I CXL) for C (z, t) and intensity I (z, t) is given by [44]: dC/dt = -(aqg) IC and dI/dz = - AI; where A (z, t) is an absorption coefficient given by A (z, t) =2.3[(a'-b) C (z, t) +bC0 F(z)+ Q]; with a', b, Q are the absorption constant of RF, photolysis product and stroma (without RF); q is the quantum yield (of RF triplet state) and a=83.6a'w, with w being the UV light wavelength. We have introduced a new rate constant (g), having a value about 0.05, such that the crosslinking time is comparable to clinical measurements. Our previous model used g=1.0 causing the crosslink about 20 times shorter than expected [44-49]. Using an effective or averaged time-independent A(z), solving the coupled equation to obtain I(z)=I0 G(z) exp[-Az], with G(z)=1- 0.25z/D; and C (z, t) =C0 F(z) exp[-aqgI(z)t]. For type-I CXL, the efficacy Ceff = 1-exp(-S), with the S function given by the time integral of [aqKIC]1/2, we obtain [46]: S=[4KCoF/(aqgI)]1/2 [1 - exp (-0.5aqgIt)] which has a steady-state formula given in Tables 1 and 2, when aqgIt>>1.
The crosslink depth (z) and time (T*) may be derived from the definition when the RF concentration is depleted to 1/ e4=0.018. We obtain the formulas given in Table 2 [44,47]: z=(1/A) ln[0.25aqgE0]; and T*=T0 exp(Az), with T0 being the surface (at z=0) cross-link time given by T0=4/(aqgI0). For q=0.5, g=0.05, T0=257/ I0 (in seconds), since a=0.62, for a'=204 (1/%/cm). For z=250 um, with a mean A=60 (1/cm), we obtain T*=257x4.5T0=1157/ I0. Therefore, for I0= (1.5, 3,9,18, 30,45) mW/cm2, in Dresden and accelerated CXL, T0= (172, 86, 29, 14, 8.6, 5.7) s; which give T*(250) = 4.5 T0= (12.8, 6.4, 2.1, 1.1, 0.64, 0.07) minutes, and T*(400) = 11.1 T0= (31.6,15.8,5.3,2.6,1.6,0.15) minutes, at z=250 um and 400 um, respectively, for cross-linked anterior and posterior stroma. In my proposed concentration-controlled method (CCM) [51], the Fdrop is defined by considering both T* (at 250 and 400 um) and the S-value on surface (at z=0), such that accelerated CXL has comparable efficacy to Dresden (at 3 mW/cm2). Table 1 shows the summary of factors influencing the CXL afficacy; and Table 2 summarizes the formulas of CXL developed by Lin [44-52].

Results and Discussions

Riboflavin dynamic profile


As shown by Figure 2, depletion of RF concentration, C (z, t), starts from the corneal surface, and gradually into the volume (z>0) given by the crosslink time T* formula which shows an exponential increasing of z. For example, for I0=10 mW/cm2, and g=0.05 (1/s), q=0.5, T0=257/I0, which increases to T*=25.7x4.5=116s, and T*=285 s at z=250 um and 400 um, respectively, for A=60(1/cm). These profile peaks are due to the non-flat initial distribution profile, F(z)<1. The strong depletion of C (z, t) will also affect the time-dependent profiles of the intensity, I (z, t), which in general will not follow the conventional Beer-Lambert law (BLL), and should be governed by the generalized, time-dependent BLL first proposed by Lin [44,46], I (z, t) = I0exp[-A(t)z], with A(t)z is the z-integration of 2.3[(a'-b) C (z, t) F+bC0F+Q], which can be numerically fit to the steady-state, A(z)=2.3[1.5bC0F+Q], when C (z, t) <t) is about 45 to 79 (1/cm), depending the depletion level of C (z, t); A1(at t=0)= 2.3[a'C0F+Q] and A2=2.3[bC0F+Q], at steady-state; nad has a mean value of A'=2.3[0.5(a'+b) C0F+Q]. The available parameters are: a'=204 (1/cm/%), Q=13.9 (1/cm), and b has a range of 50 to 100 (1/cm/%) [44].

A generalized BLL is proposed as follows to include the depletion of C(z,t) causing the increasing of the dynamic intensity:



which reduces to the initial value A1 (at t=0) and steady-state A2, when exp(-Bt)=0; with B=aqgI0exp(-A'z), A' is the mean value A'=0.5( A1+A2). Typical values are A1=79 (1/cm), A2=43.5(1/ cm), and A'=61(1/cm). We have also developed a fit value A(fit)= 2.3[mbC0F+Q], with m=1.5, when b=50 (1/cm/%), fitiing numerical simulation of the efficacy [49].
Therefore, the previous modeling [39-43] assuming a constant A(z) (about 51(1/cm), with b=0, and a flat profile (with F(z)=1), will cause a lot of errors to calculate the profiles of C (z, t), I (z, t) and thus the predicted CXL efficacy profiles. In the modeling of Semchishen, et al. they have included the RF depletion (and solved for the exact solutions), but was oversimplified by b=Q=0, and a flat profile with F(z)=1, which could underestimated about 40% of the A (z, t) value. Therefore, their optimal features and efficacy profiles are significantly different from our more accurate model, without these assumptions. The results predicted by Schumacher, et al. [40] and Kling [43] maybe correct, but limited to type-II CXL with neglected RF depletion, and an idea case that the RF has a flat profile, and only when the conventional BLL is valid. The RF dynamic profiles (with peak values), predicted by my new theory, were not found in previous modeling.

Safety criteria

The conventional accepted safety criteria for minimum corneal thickness z*=400 um, for a dose of 5.4 J/cm2, proposed by Wollensak, et al. [53,54] was meaningless without specifying the RF concentration and its profile. In addition, in their Dresden protocol, RF concentration was almost in its saturation state due to the extra instillation of RF drops during the UV exposure, in which the effective dose is about 15% less than the apparent value 5.4 J/cm2 [45,46]. Moreover, the Ed value, 0.63 J/cm2, and the threshold intensity, 0.65mW/cm2, proposed by Wollensak, et al. was also underestimated at least a factor of 5 per recent clinical data of Mooren, et al. [55]. We will analyze above issue as follows.
Formula for the the minimum corneal thickness given by a Z*- formula [46,49]: z*=(1/A') ln(E*/Ed), for the case of flat RF profile (or when D>>500 um, or G(z)=1). In general, z* is a decreasing function of C0 and D. The assumption b= 0 made by Caruso, et al. [43] will underestimate the value of A'(z) by 30% to 40%. They also assumed a flat RF profile (or G(z)=1). Hence the safety dose (E*) and minimum corneal thickness (z*) defined by their (Figure 1) is not accurate.
Defing a normalized safety dose N=(E*/Ed), with Ed being the endothelium damage threshold (at z=400 um), y z*-formula shows that z* is a nonlinear increasing function of N, but a decreasing function of RF concentration. For example: for N=6.4, we obtain z*= (475, 400, 300, 225) um, for C0= (0.05, 0.1, 0.2, 0.3) %, much smaller than that of Caruso et al z*= (350 to 540) um. For lower dose, with N= (3.2, 4.5), thin cornea of z*=250 um is still safe, for C0= (0.1, 0.2) %. These thin cornea conditions are consistent with that of Kling and Hafezi [42] and Mooren, et al. [55]. It should be noted that the z* formula is a universal function of the ratio E*/Ed, the safety dose E* depends on the accurate value of Ed, which was underestimated in animal study [53,54].
The above examples and our formulas show that the safety dose (E*) and cornea minimum thickness (z*) are determined by the collective parameters of [a,b,Q,C0,D,Ed]. Therefore, the conventional CXL safety criterion of [5.4 J/cm2, 400 um] without specifying these collective parameters, are meaningless. Furthermore, previously calculations based on Lambert-Beer law [39-42] with a time-independent A-value (of constant RF concentration) may result in errors over 30%, particularly in the steady state, which has a value about 45 to 60 cm -1, much smaller than the initial value (about 79 cm -1).

Efficacy profiles of type-I CXL

Our calculations showed that the transient profile of S function has optimal value given by when 0.5Bt=1.25, which also defines the optimal z*. It also shows the crosslinking depth(z*) is an increasing function of the dose. Figure 3 shows the steady-state profiles of the CXL efficacy which are increasing function of the diffusion depth (D); for (A) D=300 um, and (B) D=500 um; for I0 = (3,9,18,30) mW/cm2 with the same dose 5.4 J/cm2. These profiles also demonstrate that (for the same dose), lower intensity (or the SD protocol) has higher efficacy than the AC protocol (with high intensities), in consistent with recent clinical data of Choi et al. [22] and Moramarco, et al. [26]. Kling, et al. [23] recently reported the use of 1.5 mW/cm2 intensity for 30 minutes' exposure (or 2.7 J/cm2 dose) has similar efficacy as that of 3 mW/cm2 and 30 minutes exposure (5.4 J/cm2dose). This feature may be easily realized by our S-function which has an optimal dose predicted to be about 3 to 4J/cm2 [47], and the 5.4 J/cm2 (for 3 mW/cm2) is certainly higher than the optimal value.
As shown by our S-formula, it has steady state value inverse proportional to the UV light intensity (for a give dose). In comparison (to be shown later), type-II efficacy is proportional to the light dose and does not show an optimal depth z*. The above new features predicted by our modeling could not be predicted by other modeling assuming no RF depletion and a flat initial RF profile [39-43].

Nonlinear scaling law

As predicted by our S-formula and (Figure 3), the CXL efficacy at transient state (for small dose) is proportional to tI0 0.5, however,
at steady-state, it is a nonlinear increasing function of [C0/I0] 0.5 or [t/E0]0.5. This nonlinear scaling law predicts the clinical data [7-9] more accurate than the linear theory of Bunsen Roscoe law (BRL) [3,7] The conventional accelerated CXL protocol based on BRL, therefore, has undervalued the exposure time (t) for higher intensity using the linear scaling of t = [ E0 /I0], rather than t = [ E0 /I0]0.5, based on our nonlinear law. To achieve the same CXL efficacy, higher RF concentration requires higher UV light intensity; and for the same dose, higher UV light intensity requires a longer exposure time.
The BRL is based on the conventional Beer-Lambert law for UV light intensity without RF depletion, such that I(z) is time-independent, and C(z,t)=constant=C0F, therefore, S=aqKC(z)E which is a linear function of the dose E = (tI). In comparison, using a time-dependent "generalized" Beer-Lambert law [44-47], and C(z,t)= C0Fexp(-Bt), we obtain S=√(4KCoF/(aqIo) exp⁡(Az))  [1-exp⁡(-0.5aqE)]
which is a nonlinear function of E given by its Taylor expansion S=√((aqIoKCoF) exp⁡(-Az))  t[1-0.5aqE+⋯], reduces to BRL for small time with only the first term kept.Moreover, type-II efficacy, given by the time integral of [IC] (to be shown more later) follows the BRL, only when C(z) is a constant.
Our nonlinear law predicts that high UV intensity requires longer exposure time than what is calculated based on BRL. In addition, for the same dose, higher intensity depletes the RF faster and reach a lower steady-state efficacy than that of lower intensity, consistent with the recent animal data [7-9,22,23,26]. Further discussion will be shown later. As shown by our S-formula, diffusion depth (D) also plays important role. Larger D will achieve higher efficacy as shown by the RF distribution function, F(z)=1.0.5z/D, which is an increasing function of D, and F=1.0 for the idea flat distribution case.

Comparing accelerated and dresden protocols

CXL efficacy is influenced by multiple factors including, the UV light intensity, exposure period and dose, the initial concentration profiles of RF and oxygen, the quantum yield of the RF triplet state, the kinetic rate constants of RF (in type-I) and oxygen (in type-II). Besides, the protocol procedures defining how the RF drops are applied pre-operatively and during the UV exposure are also important, because they define the initial, and intraprocedure RF concentration profiles (or diffusion depth). For example, the frequency of RF drops (Fdrop) applied on the cornea after the UV is turned on, and the waiting period (Twait) for each RF drops instillation during the UV exposure. In the conventional Dresden protocol, Fdrop is about 5 to 10 times and Twait=0. In contrast, our proposed concentration-controlled method (CCM) uses Fdrop is about 1 to 3 times (for RF replenishment) and Twait is 1 or 2 minutes (for enough diffusion depth, with D>150 um).

Webb, et al. [8] recently report the measured data using Brillouin microscopy for a depth-dependent analysis after CXL. They confirmed the decrease accelerated CXL efficacy was primarily due to the lack of stiffening deeper in the cornea compared with what occurs with the standard (Dresden, 3 mW/cm2) protocol. The cutoff maximum intensity for CXL efficacy were reported as 18 mW/cm2 by Hammer, et al. [8], 34 mW/cm2 by Webb, et al. [9], and 50 mW/cm2 by Wernli, et al. [7]. Unlike findings of Hammer, et al. and Webb, et al, the study of Wernli, et al. found a relatively constant effect of CXL from 3 m W/cm2 through roughly 45mW/cm2. Webb, et al. [8] interpreted the differences in results could depend on the variability in experimental procedures. Wernli et al kept the corneas immersed in a pool of RF solution for 30 minutes before UVA exposure, whereas in the protocols of Webb et al and Hammer, et al. RF drops were incrementally applied to the cornea. This factor may affect the efficacy, however I believe, it would be a minor factor, if comparable diffusion depth is achieved in both protocols (after 30 minutes waiting period). Other factors such as the frequency of RF drops applied and their waiting period (for enough diffusion) during the UV exposure, could play a more important role. Greater details are discussed in the following.
Kling, et al. [23] recently reported the use of 1.5 mW/cm2 intensity for 30 minutes exposure (or 2.7 J/cm2 dose) has similar efficacy as that of 3 mW/cm2 and 30 minutes exposure (5.4 J/cm2 dose). This feature may be easily realized by our S-function which has an optimal dose predicted to be about 3 to 4 J/cm2 [47], and the 5.4 J/cm2 (for 3 mW/cm2) is certainly higher than the optimal value.

Cut-off maximum intensity

Validation of BRL for accelerated CXL has been studied by Wernli, et al. [7] by the Cutoff maximum intensity about 50 mW/ cm2 and a minimum crosslinking time about 2 minutes. These criteria may be derived by our S-function as follow. Taking a threshold value of S0 (the minimum S for efficient crosslinking as that of Dresden 3 mW/cm2), or 4KC0Fexp(Az)/(aqKI0) > S0 2, from our S-formula, which leads to a cutoff maximum intensity (on the corneal surface, z=0) given by I*=4KC0/(aqKS*2). For C0=0.1%, q=0.5, K=7.8, K'=0.05, a=0.622, we obtain I*=201/S0 2, or I*= (50.3,22.3) mW/cm2, for S0= (2,3), i.e., Ceff=1.exp(-S0) = (0.86,0.95). These values predict what was reported by Wernli, et al. [7] and Hammer, et al. [8], but higher than Webb, et al. [9]. For the maximum intensity, the associate minimum exposure time is given by Tm=E0/I*= (1.8, 0.8) minutes, (for E0=5.4 J/cm2). We should note that the S-formula is valid for the situation of non-controlled RF concentration, i.e., no extra RF drops were applied during the UV exposure (or Fdrop=0). A concentrationcontrolled methods (with Fdrop=1 to 3) will be discussed later to overcome the limitation of maximum intensity.

Demarcation-line depth

The definition of crosslink depth (z) was defined earlier by when the RF concentration is depleted to 1/e4=0.018 giving the formulas given in Table 2. z=(1/A) ln[0.25aqgE0]. Alternatively, it maybe also defined by when the CXL efficacy is larger than a threshold percentage, or at the peak of S-profiles shown by Figure 4. Demarcation line (DL) depth is widely used as an indirect marker of CXL efficacy. The DL depth after AC CXL appears to be shallower than that of conventional SD protocol, thus less corneal volume seems to be cross-linked with the AC protocol [26-30]. The CXL efficacy was theoretically proposed [49,51] and proportional to the cross-linked corneal stroma volume (V), defined by both cross-link depth (z) and cross-link strength (S), i.e., V=zS. For the same dose, similar z can be achieved by AC and SD, but the cross-linked-S in AC is always smaller than that of SD as shown in Figure 2, thus AC has smaller cross-linked V and efficacy. AC method achieves a lower steady state S-value due to the faster (stronger) RF depletion, thus they reach a lower steadystate (saturated) value than SD (using lower intensity). As shown in Figure 4, the calculated CXL steady-state efficacy and the DL depth (based on measured data [29]) follow the similar decreasing trend in accelerated CXL. However, their functional dependences on the UV light intensity (or exposure duration) are very different. Toker, et al. proposed a fit formula for the DL depth, Z= 960 exp (t/79) - 6138 exp (-I0) - 831. We propose another fit formula based on intensity (for a fixed dose): Z= 600[1- 0.48exp (0.012I0)], with Z in um and I0 in mW/cm2. In comparison, our modeling provides the CXL efficacy, Ceff = 1- exp [-0.116 (E0/I0)0.5] or 1- exp (-0.116 t 0.5), for a fixed dose E0 = tI0. From the above comparisons shown by Figure 4, it is reasonable to conclude that the CXL efficacy (Ceff) is linearly related to the DL depth for a UV intensity up to about 30mW/cm2. However, for high intensity range (>35 mW/cm2), the suddendrop of DL depth is stronger than Ceff. Similarly, Wernli, et al. also reported the sudden-decrease of Ceff for high inentisty (>45 mW/cm2), which maybe better fit to our calculated Ceff, if we include the additional influencing factor of extra RF drops during the UV exposure to be described as follows.

Concentration-controlled protocol

To overcome the drawback of low efficacy in accelerated CXL as predicted by the S-formula, a RF concentration-controlled method (CCM) is proposed [51] as follows. In the conventional Dresden protocol, extra RF drops were instilled during the UV exposure (with a frequency Fdrop=5 to 10) which reduced the effective dose from 5.4 J/cm2 to about 4.0 J/cm2, based on our calculations [47,49].

For an optimal protocol (for fast and efficient CXL in the anterior stroma), I previously proposed Fdrop =1 to 4 to compensate the fast RF depletion in the anterior stroma, specially in high intensity (>18 mW/cm2). 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 a period (t*) about 1.0 to 2.0 minutes to allow enough RF diffusion (with a diffusion depth D>150 um) before it is turned on again. We note that the Fdrop is proposed by the combined consideration of the crosslink time (T*), which defines when RF is depleted; and the surface S-value, which defines the strength of CXL, or the number of re-applying RF drops needed to achieve the same S-value for all intensity ranges (1.5 to 45 mW/cm2). In the above proposed CCM, my theory predicts the comparable efficacy (for the same dose) for intensity of 1.5 to 45 mW/cm2, 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 [I0/3]xN, N being the Fdrop for optimal efficacy at our referenced intensity (3 mW/cm2) [51].

The Fdrop of CCM is formulated based on two revisions on BRL for the total UV exposure time (t): (a) the effective dose, a fraction (R) of Dresden, and (b) the extended exposure time (or dose) based on Lin's Nonlinear (LN) scaling law t0.5, or E0/[I0]0.5. Therefore, t(LN)=Rt(BRL)[I0]0.5 = R(E0/I0)[I0]0.5= R[E0/[I0]0.5. If we choose an effective dose 4.0 J/cm2, R=4.0/5.4=0.74, then t(LN) = (31,22,12.6,9,7,5.7) minutes, comparing to t(BRL) = (60,30,10,5,3,2) minutes, for I0= (1.5, 3,9,18, 30,45) mW/cm2. Given the crosslink time T*(400)= (31.6,15.8,5.3,2.6,1.6,1.05) minutes, we obtain (taking the closest integers) Fdrop = t(LN)/T*= (1,1,2,4,4,5).
For effective dose 2.7 J/cm2, R=0.5, t(LN) = (21,15,8.5,6,5,4) minutes, Fdrop = (1,1,2,2,3,4). Note that the 2.7 J/cm2 dose was reported by Kling, et al. [57], showing that 1.5 mW/cm2 (for t=30 minutes) has the similar efficacy as that of standard Dresden protocol using 3 mW/cm2 (for t=30 minutes) and 5.4 J/cm2 dose. Which implied that 5.4 J/cm2 dose might be overdone. In comparison, if BRL is used, Fdrop = t/T*= 2, for all intensity when same dose (E0) is used. In this case Fdrop is independent to I0, since t=E0/I0, and T*=T0/I0.
Note that t(LN) is longer than t (Dresden, based on BRL and 5.4 J/cm2).
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 to 250 um) which is retreated by Fdrop times, and the waiting period (with UV off) after each RF drops secures enough diffusing depth (D>150 um). 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. [7] than the simple protocol (with Fdrop=0) or Dresden protocol (with Fdrop >5). Figures 4 and 5 shows an example of the combined S-function (for 18 mW/cm2 intensity), with Fdrop=3, with first instillation having D1=500um, followed by D2=200um and D3=100um.

Photochemical kinetic (Type-I and -II)

The photochemical kinetics of CXL is shown in Figure 6 for both type-I and -II. Greater detailed kinetic of type-II only was published in our prior work [48], we will now present the combined kinetic. Figure 6 shows the typical depletion profile of oxygen which defines the contribution of type-I and type-II in different stage. Typical depletion time of oxygen is about 5 to 15 seconds, for light intensity of 30 to 3 mW/cm2, per measured data of Kamaev et al [39], and takes about 10 minutes for the oxygen to resupplied or replenished to about 1/3 of its initial state. Therefore, during this transient stage of 10 to 20 seconds, As shown in Figure 6, the CXL process has three pathways. Defining [S0], [S1] and [T3] are the ground state, singlet excited state, and triplet excited state of RF molecules which is excited by the UV light. In type-I pathway, T3 can interact directly with the collagen substrate (A); or with the ground state oxygen (O2) to generate a superoxide anion (O-); in type-II pathway, T3 interacts with the ground oxygen (O2) to form a reactive singlet oxygen (O*) [48]. The RF excited triplet can undergo two kinds of reactions. In type I reaction, it can react directly with a substrate (cell membrane or a molecule), and transfer a proton or an electron to form a radical anion or radical action, respectively. These radicals may further react with oxygen to produce reactive oxygen species (ROS). Alternatively, in a type II reaction, the triplet RF can transfer its energy directly to molecular oxygen (triplet in the ground state), to form excited- state singlet oxygen. As show by Figure 6, both type-I and type-II reactions can occur simultaneously, and the ratio between these processes depends on the type of photosensitizers (PS) used, the concentrations of PS, substrate and oxygen, the kinetic rates involved in the process. Referring to the three kinetic pathways of Figure 6, a set of macroscopic kinetic equation for the RF ground-state, C (z,t), and the ground state oxygen molecule [O2] (at the quasi steady-state is constructed [48,49,52].



where G(z,t)=C(z,t)[O2]G0, G0=/([O2]+k+K'), g=k8[A]G0/k3, k=k5/k3; K'=k8[A]/k3; K12= (s1k11/K1+ s2k12/K2); q is the triplet state [T] quantum yield given by q=k2/(k1+k2); s2 and s1 are the fraction of [O2] converted to the singlet oxygen and other ROS, respectively, in type-II and type-I. In Eq. (2.b) we have included a fit parameter (N=10.9) which can be fit to the measured oxygen profiles at various UV intensity [39], for [O2]=0, at about 5 to 10 seconds for intensity of 30 to 3 mW/cm2 [39]. And the UV light intensity in the corneal stroma is given by a time-dependent Beer-Lambert law [44], as shown b Eq.(1). We have also include in Eq. (2.b) the oxygen source term P(z,t)=p(1-X/X0), with a rate constant p to count for the situation when there is an external continuing supply, or nature replenishment (at a rate of p), besides the initial oxygen in the stroma.

Overall efficacy analysis (type-I and -II)


The normalized efficacy defined by Ceff =1-[A]/[A]0 =1-exp(-S), with S-function for type-I (S1) and type-II (S2) to be derived as follows. The type-I efficacy may be further expressed by rate equation of conversion of collagen monomers [M] to polymers, where the NOM term of Eq. (1.a), g=k8[A]G0/k3, is replaced by an overall rate costant (K) including all polymerization chain reactions. The more accurate S1 is given by [49,52]





where f is the fraction of ROS interacting with [A]. The first termin Eq. (5.a) relates to the direct coupling of triplet state [T] with the substrtae [A] under hypoxic conditions or any other nonoxygen- mediated (NOM) reactions; and the second term relates with the (ROS)-mediated reactions (in type-I). f is the fraction of all ROS (including singlet oxygen) interacting with acceptors [A], or the oxygen-mediated (NOM) reactions in type-I and type-II. S s2 and s1 are the farction of [O2] interacting with [T] to produce singlet oxygen (in type-II) and other ROS (in type-I), respectively. The overall CXL-efficacy is given by Ceff=0.5[CX1 + CX2], with CX1=1-exp(-S1), CX2=1-exp(-S2).

Coupling between type-I and type-II

As shown by Eq. (2), the coupling of OM and NOM, or type-I and -II are given by various factors: the depletion of RF concentration, C (z,t), and the efficacy S functions both are function of G0, where G0 is governed by both OM (in type-I) and ROS-mediated (in type-I) terms and the NOM term also affect the solution of C(z,t) in Eq. (1.a). The strong coupling between OM and NOM resulted from K12G (z,t) which is given by K12G= K12C[O2]/([O2]+k), with K12= (s1k11/K1+ s2k12/K2) having the OM terms of type-I and -II. The NOM term of type-I also shown in k= (k5+ k8[A])/k3 which involved in G(z,t). For predominant type-I process, the RF depletion due to ROS-mediated term K12G [O2] is minor and can be ignored in Eq. (2.a). However, it affects the G function and the type-II S-function as shown by Eq. (2.b). According to the proposed mechanism of Kamaev, et al. [39], under aerobic conditions, they believe that CXL in the cornea is initiated due to the direct interaction between the substrate and excited RF triplets, with singlet oxygen playing a limited and transient role in the process. In contrary, Kling, et al. [42] believed that type-II is the predominant mechanism. As show by Figure 6 and 7, our new modeling system demonstrated theoretically that CXL using RF as the PS is predominated by the NOM term of type-I, or the direct coupling of triplet RF to the substrate [A], since the OM pathways (in both type-I and II) via singlet oxygen playing a limited and transient role in the process per Kamaev, et al. [39], who proposed the mechanisms but did not developed the detailed macroscopic equations shown in this study.
Sufficient external oxygen supply, either pre-CXL or during CXL, may enhance the CXL overall efficacy, but it is still limited to its transient state. Clinically, UV light in pulsing-mode may also improve the efficacy, but only when enough UV-off period is available, about few minutes, for oxygen replenishment. Thin cornea [56] with low UV intensity (3 to 18 mW/cm2), and epi-off CXL will achieve higher overall efficacy than that of thick corneas, or epi-on CXL under high intensity (>18mW/cm2) in its transient state (Figure 7).

Conclusion

The debating issues are solved/analyzed mathematically in this study include: the validation of BRL and BLL, the intensity cutoff maximum of accelerated CXL, the safety criteria, the demarcation depth, the role of oxygen in both type-I and type-II CXL. CXL in the cornea is initiated due to the direct interaction between the substrate and excited RF triplets, with singlet oxygen playing a limited and transient role in the process. The overall CXL efficacy, is proportional to the UV light dose (or fluence), the RF and oxygen initial concentration and their diffusion depths. A new protocol using concentration-controlled method (CCM) can improve the efficacy in accelerated CX, which is less efficient than the Dresden (low intensity) CXL under the normal, noncontrolled methods.

Competing interests

The author is the CEO of New Vision Inc. and has financial interest.
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Tables & Figures


Figure 1: Photochemical kinetic pathways of CXL, including type-I and type-II mechanisms [48].



Figure 2: The normalized RF concentration profile at exposure time t= (0, 12, 25, 50, 115) seconds (for curves 1,2,3,4,5), for diffusion depth D=500 um, initial C0 =0.1% and intensity I0 = 10 mW/cm2, where red-line indicates RF depletion level of 0.018 [44,45].



Figure 3: Steady-state profiles of the CXL efficacy for (A) D=300 um, and (B) D=500 um; for I0 = (3,9,18,30) mW/cm2, for curves (1,2,3,4), with same dose of 5.4 J/cm2 and C0 = 01. % [47,49].



Figure 4: The CXL efficacy (red dots, calculated) and demarcation line depth (blue bars, measured data also fit to an exponential function) vs. UV light intensity showing the nonlinear decreasing features.



Figure 5: The combined S-function, for Fdrop=3, where top curve is the combined function of curve1,2,and 3 showing a high efficacy than curve 1 (with Fdrop=0).



Figure 6: The kinetics of CXL, where [S0], [S1] and [T3] are the ground state, singlet excited state, and triplet excited state of RF molecules (see text for more details)[48].



Figure 7: Schematics of the oxygen profiles during the CXL process; in the transient stage, both type-I and –II coexist until the oxygen is depleted; then type-I dominates before the oxygen is resupplied or replenished.



Table 1: Critical factors influencing the safety and efficacy of CXL [49,52].



Table 2: Summary of CXL formulas referred to Table 1 for more details and definitions of parameters [44,47-49].

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