Combined Analysis of Safety and Optimal Efficacy in UV-light-activated Corneal Collagen Crosslinking

Aims: To analyze the combined factors for safety and optimal efficacy for UV-light-initiated corneal collagen cross-linking (CXL). Study Design: Modeling and analysis of CXL. Place and Duration of Study: New Vision Inc, Taipei, between Oct. 2015 and July 2016. Methodology: Analytic formulas were derived based on the coupled dynamic equations for the safety dose (E*), minimum corneal thickness and concentration of the riboflavin, and the cornea stiffness increase after the CXL. The critical parameters influencing the efficacy of CXL include: various absorption coefficients, initial concentration (C0) and diffusion depth (D) of the riboflavin solution, the quantum yield, the UV light intensity (I0) and dose (E), irradiation duration (t), the cytotoxic threshold dose of endothelial cells(E’) and the corneal thickness (z). Conclusion: The safety dose (E*) is an increasing function of the parameter set (D, z, C0) and has a range of 5.3 to 10.1 J/cm 2 for cytotoxicity threshold 0.63 to 1.26 J/cm 2 . Minimum corneal thickness z* =(300, 400) um for dose of E*=(5.0, 10.1) J/cm 2 which has a much wider range than the conventional safety dose 5.4 J/cm 2 (with z*=400 um). For maximum efficacy, the optimal dose is 0.7 to 1.55 J/cm 2 . However, to achieve crosslink depth of 230 to 300 um, higher dose of 2.0 to 3.0 J/cm 2 is recommended. Original Research Article Lin; OR, 6(2): 1-14, 2016; Article no.OR.28712 2


INTRODUCTION
Photopolymerization has been widely used in many applications ranging from chemical engineering to biomedical engineering and biomaterials [1][2][3][4].It has been used for production of catheters, hearing aids, surgical masks, medical filters, and blood analysis sensors [2].Photopolymers have also been explored for uses in dentistry, drug delivery, tissue engineering and cell encapsulation systems [3].Both ultraviolet, visible and infrared lights (360 to 1000 nm) have been used as the photoinitiators for various photosensitizers [3,4].
In CXL, both type-I and type-II photochemical reactions occur [11], The photosensitizer riboflavin absorbs the UV energy and is excited to its triplet state (3Rf*).In type-I mechanism, the excited triplet state reacts directly with the collagen proteins and creating substrate free radicals or radical ions of superoxide anion (O 2 -) and OH, respectively, by hydrogen atoms or electron transfer and the sensitizer is consumed during the reaction.In type-II mechanism, the excited sensitizer reacts with oxygen to form reactive singlet oxygen (RSO), 1 O 2 and regenerates ground state sensitizer.These RSO then further reacts with the collagen covalent bonds between the collagen molecules and proteoglycans to produce additional cross-linked bonds [11].Unlike type-I process, type II process does not consume sensitizers during the reaction.but requires oxygen and therefore it is the minor mechanism in most CXL processes.CXL can harden the collagen but also damage viable cells when the UV dose is higher than the damage threshold or when the antioxidant defense system is overwhelmed.
It has been reported that oxygen concentration in the cornea is modulated by UV irradiance and temperature and quickly decreases at the beginning of UV light exposure [11].The oxygen concentration tends to deplete within about 10-15 seconds for irradiance of 3 mW/cm 2 and within about 3-5 seconds for irradiance of 30 mW/cm 2 [11].By using pulsed UV light of a specific duty cycle, frequency, and irradiance, input from both Type I and Type II photochemical kinetic mechanisms may be optimized to achieve the greatest amount of photochemical efficiency.The rate of reactions may either be increased or decreased by regulating one of the parameters such as the irradiance, the dose, the on/off duty cycle, riboflavin concentration, soak time, and others [11,34,37].

This
study will demonstrate that the conventionally accepted criterion having a safety dose E*=5.4 J/cm 2 and a corneal minimum thickness z*>400 µm is just one of the special case meeting our safety criteria.Without specifying the parameters of riboflavin (RF) concentration and its diffusion depth, the conventional safety criteria (E*, z*) = (5.4J/cm 2 , >400 µm) is meaningless.Furthermore, the safety dose E* is proportional to the cytotoxic threshold of endothelial cells.Therefore, the accurate safety dose relies upon the accurate threshold dose which was estimated (or proposed) to be 0.63 J/cm 2 by prior work based on the measured animal data [25,26].Moreover, the conventionally accepted UV light absorption in RF solution [22,23] is based only on its initial (at t=0) intensity and ignoring the RF dynamic depletion causing the steady-state intensity higher than the initial value.Therefore, the prior work [22,24] overestimates the dynamic light intensity.
A recent article of Mooren et al [40] evaluated the cytotoxicity threshold for human endothelium with setting from the endothelial side and concluded a much higher value (at least eight times higher) than that of the previously reported animal models (0.63 J/cm 2 ) [25,26].Based on their measured human data, the corresponding safety surface dose (based on our calculation) is as high as 27 J/cm 2 .This value is much higher than any of the clinically reported data [21][22][23][24][25][26].Therefore further validation of the cytotoxicity threshold for human endothelium is required.Furthermore, it was reported by Lombardo et al. [40] that thin cornea with 325 um thickness after the CXL did not shown endothelium damage under a high dose of 5.4 J/cm 2 .These reported data indicated that the conventional safety dose [25,26] 5.4 J/cm 2 and minimum corneal thickness of 400 um are underestimated.This study will demonstrate the new finding that thin human cornea of 300um (much less than the conventional animal model of 400 um) is stilled allowed in CXL-Lasik process for endothelium threshold dose higher than 1.8 J/cm 2 .Our new theory will show that the safety dose (E*) and the corneal safety thickness (z*) cannot be set as a constant as that of conventional protocols.Instead, they are variable functions defined by the combined parameter set.This study will use the currently available (or measured) data in the calculations, whereas the unknown parameters will be treated as free parameters within the clinically recognized ranges.This study will focus on the analytic formulas, whereas numerical results based on computer simulations will be shown else where.

The Modeling System
As shown in Fig. 1, a corneal model consists of its epithelial layer and the underlying stroma collagen.The UV light is normal incident to the corneal surface.A typical CXL protocol is to administer RF on the corneal surface 5 to 10 times at 2 to 5 minute intervals and wait until the RF solution diffuses into the top layer at approximately 200 to 300 µm.The CXL procedures could be conduced (as shown by Fig. 1) either with epithelium removed (epi-off) with a 0.1% riboflavin-dextran solution or with epithelium intact (epi-on) with a 0.25% riboflavin aqueous solution.It was known that riboflavin diffusion depth in the epi-on case is normally less than that of epi-off and therefore the epi-on is less efficient [5].In the above described CXL model, the UV light intensity in the corneal stroma is given by a revised time-dependent Lambert-Beer law [18].
where the time-dependent extinction coefficient A(t) shows the dynamic feature of the UV light absorption due to the RF concentration depletion.Without the RF, A(t) becomes a constant given by the absorption coefficient of the corneal stroma tissue reported to be 27 A=2.3Q, with Q=13.9 (1/cm).With the RF in the stroma, the initial (at t=0) overall absorption has an extra absorption defined by the extinction coefficient and initial concentration of the RF, i.e.A (t=0) =2.3 (Q + 1 ε C 0 ), with the reported data [20,28] 1 ε = 204 (%•cm) -1 .For t>0, A(t) is an increasing function due to the deletion of C 0 in time and defined by both the extinction coefficient of the RF( 1 ε ) and its photolysis product ( 2 ε ), where 2 ε is not yet available for human, but was estimated to be about 80 to 120

UV light UV light
Cornea Cornea epi (%•cm) -1 , based on measured data in RF solution under UV light irradiation [18].
In the CXL process, the UV light intensity ( , ) I z t and RF concentration C(z,t) in the corneal stroma may be described by a set of coupled first-order differential equations, or by the integral equations [16][17][18][19][20].
(2.a) ε are the extinction coefficients of RF and the photolysis product, respectively.I 0 is the initial UV light surface intensity, or I(z=t=0)=I 0 .; and C 0 is the initial RF surface concentration assuming a distribution profile given by F(z) = 1 -0.5z/D, or C(z,t=0)=C 0 F(z), with a diffusion depth D in the stroma.
The prior work of Schumacher et al [22] based on a non-depleted RF concentration, i.e, the assumption of aE=0 in Eq. (2.b), significantly overestimates the RF concentration for t>0.Moreover, the time-dependent of I(z,t) and C(z,t) in their Eq.( 5) and ( 6) is the pre-treatment time, rather than the actual UV light exposure time.That is, after the pre-treatment time (t'), they treated the CXL as a steady process without solving the dynamics of CXL.Their t' corresponding to our initial time (t=0).The profiles shown in their Fig. 1 are just the initial (at t=0) profiles of our Eq.(2).The prior work of Schumacher et al [22] also used an oversimplified model to assume 2 ε =0.Therefore, their calculated profiles, Fig. 3 and 4, significantly deviate from our exact numerical profiles to be shown later.
The reported measurements [27,28] provide the parameters of 1 ε = 204 (%•cm) -1 and Q = 13.9 (cm -1 ), whereas 2 ε is not yet available in human, but was estimated to be in the range of 80 to 120 (%•cm) -1 by the RF depletion test [17,18] and the quantum yield φ will be a free parameter In Eq. (2), the following units are used: C(z,t) in weight percent (%), I(z,t) in (mW/cm 2 ), λ in cm, Q in (cm) -1 and j ε (for j=1,2) in (%•cm) -1 .As shown by Eq. (2) that there are three major UV absorption components in the CXL process: the absorption of the stroma tissue (Q), which is independent to the RF concentration; the absorption of the unreacted RF solution ( 1 and the photolysis product ( 2 ε C 0 ), both are proportional to the initial RF concentration C 0.
( ) and the steady state light intensity is derived by using ) where (for j=1,2) where A 1 (for j=1) is the initial state (at t=0) absorption coefficient, independent to 2 ε ; and A 2 (for j=2) is the steady-state absorption coefficient, which is independent to 1 ε because of the complete concentration depletion, ( , ) 0 We note that G(z) in Eq. (3.d) is the integration of F(z) over z.

The Effective and Safety Dose
The effective dose (or fluence) applied to the cornea collagen (at a depth z) for a UV exposure time (t) is defined by [20].
The UV light intensity is a dynamic function of time and z and requires a full numerical (2).For comprehensive analytic formulas, various numerically fit techniques were presented earlier 18-20 .In this study we will use the linear approximation of the light intensity I (z,t) given by the mean value of the initial (I 1 ) and steady intensity (I 2 ) defined by ( , ) 0.5 ( ) ( ) for steady-state, or t>T, where T is the steady-state cross-linking time given by T=m/(aI 0 ), with m =12 fit numerically with the exact solution of Eq. ( 2) to be shown else where.Time integration of the light intensity, the effective dose (at z) Eq. ( 4) becomes [20].
where E =I 0 t is the surface dose (at z=0), and g is a correction factor for the transient state given by g= 0.5(I 2 -I 1 )(T/E),T=m/(aI 0 ), with fit parameter m =12.For ) and C 0 = 0.1%, we obtain an approximated value of the correction factor g=0.97[exp(-A 2 z) -exp(-A 1 z)], which is about 15% correction to the steady-state when E= 3.0 to 4.0 J/cm 2 and at z=400 um.
Given the cytotoxicity threshold dose of the endothelium Eeff=E'(at a depth z), the surface safety dose (defined on the corneal surface z=0) is given by E=E* in Eq. ( 5) to obtain an E*formula [20].
For a given E' and E 0 , the normalized safety dose, E*/E 0 is given as follows.
The safety dose (E*) is defined by the maximum dose (on the cornea surface, z=0) without causing the endothelial cells damage (at a depth of z=400 µm), which is given by the measured dose in the animal model [24][25][26], 3 mW/cm 2 x60x30 seconds=5.4J/cm 2 .We will show later that the UV intensity at the endothelial cells (at a depth of z=400 µm) estimated as 0.35 mW/cm 2 and the threshold dose 0.35 mW/cm 2 x60x30=0.63J/cm 2 are under estimated based on the simplified model [22] assuming a constant RF concentration, or aE=0 in Eq. (2).For human cornea, the damage threshold dose of the endothelial cells were reported to be much higher than the animal model [41].

The Minimum Thickness and Concentration
Solving Eq. (7.a) for z=z*, using a mean value of G(z=400 um)=0.8, and B<<1 is neglected, we obtain the safety (or minimum) corneal thickness as follows.

The Efficacy Profiles
The kinetic equation of the rate of polymerization is mainly determined by the rate of reacting monomers given by [6, 16,20].
Above equation shows that photoinitiation rate (R) is a product of two competing factors, the RF concentration and the laser intensity.Therefore, an optimal value of Using the UV light intensity defined by the mean value Eq. (3) and C(z,t) given by Eq. (8), we obtain analytic formula for the photoinitiation rate (R).
where H(z) is given by the mean intensity H(z)=0.5(I 1 + I 2 )/I 0 , and E(z) is given by an analytic formula, Eq.
These optimal features are due to the competing between UV light intensity and RF concentration which have opposite trend.
Integration of Eq. (13) over time (t), and using Eq. ( 5) for E(z), we obtain the profile (z-dependence) of the increase in corneal stiffness (S').
The efficacy of CXL may be defined by the increase in corneal stiffness (S) after CXL given by the total amount of induced crosslink.Integration of Eq. ( 14) over z, and taking averaged over the CXL depth (z), we obtain the normalized increase in stiffness (S).

The Scaling Laws
From E. (14), for small (atI 0 )<<1, the transient state of S'(z,t) is proportional to (atI 0 ), we obtain the scaling law . ( which shows that the transient state of S' is linearly proportional to the exposure time (t) for a given UV intensity (I 0 ), and is proportional to the square root of I 0 for a given t.However, the steady-state, when (atI 0 )>>1, exp(-0.5atI0 )=0 in Eq. ( 15), has a different scaling law given by The above steady state is proportional to I 0 -0.5 , comparing to the transient state S is proportional . The opposite trend is due to the time integration over time (t) which causes the dependence of S' as I 0 -0.5.These features will be reconfirmed by our numerical data later.
The above scaling laws based on the approximated Eq. ( 13) will be used to analyze the exact numerical results be shown elsewhere.The optimal C 0 * and I 0 * and the scaling laws for S(z,t) may be mathematically derived similar to that of S'(z,t).However, analytic formulas are not available.

RESULTS AND DISCUSSION
Numerical calculations of the factors influencing the CXL efficacy based on Eq. ( 2) and (15) will be presented elsewhere.This paper will focus on the results based on the developed analytic equations ( 6), ( 7), ( 10), ( 11), ( 14)- (17).The major clinical issues of CXL will also be discussed.

The Effective Dose
The conventional CXL procedures are based on the Dresden protocol which requires a surface safety dose of 5.4 J/cm 2 and a minimum corneal thickness of 400 µm.During the UV exposure, riboflavin drops were applied every few minutes for saturated riboflavin concentration in the stroma and extra protection of the corneal endothelial cells.However, the effective dose of CXL is reduced by the extra absorption of this surface layer.For maximum efficacy, the wasteddose can be avoided by washing out the extra surface riboflavin layer after its sufficient diffusion into the stroma and prior to the UV light exposure.For RF surface layer (with thickness d), the energy absorbed the RF layer is given by Eab=E(t1/t)[1-Re], where t1 is the portion of the exposure period (t) having RF layer on the surface and R is calculated from Eq..(5) with Q=0, and a mean value of A=15 C 0 , Re=exp(-15dC 0 )=(0.86,0.74),for d= (100, 200) um and C 0 =0.1%,For example, for t1/t=0.7, the conventional dose 5.4 (J/cm 2 ) is reduced to an effective dose of Eeff=E-Eab=4.4and 4.9 J/cm 2 , for d=200 and 100 um. that is 10 % to 20% of the dose, or 5.4 to 1.08(J/cm 2 ) dose is wasted in the B2 surface layer having a thickness of 100 to 200 um um and concentration of 0.1%.In the current Dresden protocol (using a 5.4 dose J/cm 2 ), we estimate the effective dose has a range of 4.0 to 5.0 J/cm 2 , depending on the frequency of RF instillation during the UV exposure.

The Safety Dose
As shown by Eq. ( 6), the safety dose E(z,t) is an exponential increasing function of D, C 0 , 2 ε , Q and the corneal thickness (z), as shown by Eq. ( 7).In a recent article by Mooren et al [40], they evaluated the cytotoxicity threshold for human endothelium (E') and concluded a much higher value (at least 5.4 J/cm 2 ) than that of the previously reported animal models (0.63 J/cm 2 ).Using E'=5.4 J/cm 2 the corresponding safety surface dose, based on Eq. ( 6), gives 5.4x6.3=27J/cm 2 for C 0 =0.04 %.This value seems to be too high clinically.Therefore, the actual cytotoxic threshold of endothelial cells in human requires further studies.

The Minimum Corneal Thickness
Fig. 5 shows that for a given safety dose E*=E 0 , the safety (minimum) corneal thickness (z*) is a decreasing function of the RF concentration but is an increasing function of the diffusion depth D as also shown by Eq. ( 10).
The above examples demonstrate that thin human cornea of 300 um (much less than the conventional animal model of 400 um) is stilled allowed in CXL-Lasik process as far as the endothelium threshold dose is higher than 1.26 J/cm 2 which was reported recently [40,41].

The Stiffness Profiles and Crosslink Depth
Detailed computer simulations for the influence of each of the parameters of [C 0 , D, I 0 , t, φ ] in the increase of stiffness defined by Eq. ( 14) and (15) will be shown elsewhere.This paper is focusing on the features summarized from the numerical simulations, besides typical examples of the stiffness profiles.Fig. 7 shows the typical profiles (z-dependence) of the increase in corneal stiffness (S') for a fixed UV light intensity I 0 =10 W/cm 2 , defined by Eq. ( 14).Each profile has a maximum value S* (at the crosslink depth Z*), where both S* and Z* are increasing function of the UV exposure time (t), or dose (for a given light intensity).We note that the crosslink depth (Z*) calculated numerically by the maximum S* has the similar functional form as that of Eq. (9.b) which is analytically derived.
Fig. 8 shows S' versus z for a fixed exposure time t=100 s, but for various UV light intensity, where S* is a decreasing function of the UV light intensity (for a given exposure time), whereas Z* is an increasing function of I 0 (or dose E 0 =tI 0 ) which is shown in Fig. 9 and can be well fit by Eq.where units are S* in K 0 , E in J/cm 2 , C 0 in %, Z* and D in um.Eq. ( 18) provides the important feature that S* is an increasing function of C 0 and D, but it is a decrease function of E. In comparison, Z* is an increasing function of E and D, but it is a decrease function of C 0 .
As shown in Fig. 9, the crosslink depth is only Z*=(120 to 170) um at the optimal dose E 0 *=(0.7 to 1.3) J/cm 2 .To achieve Z* in the range of 230 to 300 um (as shown by Fig. 9), one requires higher dose of E 0 =2.0 to 3.0 J/cm 2 despite the drop of 10% to 18% from its optimal value.Numerical simulations to confirm the scaling laws shown by Eq. ( 16) will be presented elsewhere.Using a reasonable value of 2 ε =102(%•cm) -1 and Q=13.9 (cm -1 ), and the conventional dose 5.4 (J/cm 2 ) in a Dresden protocol is reduced to 4.0 to 5.0 J/cm 2 , depending on the frequency of RF instillation during the UV exposure, or the thickness of the RF surface layer (about 100 to 200 um).Therefore for efficient CXL the extra RF layer should be washed out prior to the UV exposure.We note that a wide range of the peak RF concentration (C 0 ) were reported 0.04% [40], 0.06% [41], 0.02% [42] and 0.1% [22], therefore further clinical measurements are needed for conclusive values.

The safety dose
The cytotoxicity threshold endothelium cells (E') was reported in an animal model of 0.63 J/cm 2 .[26] which is much lower than the reported human data [40] of at least 5.4 J/cm 2 .The corresponding safety surface dose (at the reference point of z=400 um and C 0 =0.1%), based on Eq. ( 6), is 5.3 J/cm 2 (using animal model) and 27 J/cm 2 using human mode which seems to be too high clinically.Therefore, the actual cytotoxic threshold of endothelial cells in human requires further studies.Knowing the actual value of E', 1

The accelerated CXL protocol
The accelerated CXL uses a higher UV intensity (for a fixed dose of 5.4 J/cm 2 ) to reduce the UV exposure time (t) based on the Bunsen-Roscoe law (BRL) that a photochemical reaction will stay constant if the total energy is constant and a shortened irradiation time at higher irradiance should lead to the same increase in biomechanical stiffness as a longer irradiation time at lower irradiance.For examples: t=(30, 10, 5, 3, 2) minutes, for UV intensity I 0 =(3, 9, 18, 30, 45) J/cm 2 based on BRL.This linear feature may be described by Eq. ( 9), where the surface crosslinking time given by T 0 =T*(z=0)=1000M/(aI 0 ) which follows the RBL.However the T* (with z>0) is a nonlinear function of I 0 and does not follow the RBL to be further analyzed as follows.

The nonlinear scaling laws
As shown by Eq. (16.b), the scaling laws predict the nonlinear features against the RB law which was also reported clinically [29].Eq. (16.b), also shown by Fig. 8, predicts the steady state maximum stiffness (or efficacy) is proportional to I 0 -0.5 , that is the UV exposure time (for a given dose) should be adjusted for longer than those based on the linear RBL.

The optimal dose
As shown by Fig. 9 and 12, to achieve Z* in the range of 220 to 300 um, one requires a dose of E 0 =2.0 to 3.0 J/cm 2 which is much lower than the conventional "appearance" dose of 5.4 J/cm 2 , but slightly lower than the effective dose of 4.4 to 4.9 J/cm 2 .(after reducing the absorption of the RF surface layer).

The role of concentration and diffusion depth
As show by Eq. ( 17), and Fig. 10, the crosslink depth (Z*) is an increasing function of the diffusion depth (D), but a decreasing function of the concentration (C 0 ).Therefore, installation of the RF solution on the epi-off corneal surface must be long enough (at least 15 minutes) to allow a deep diffusion depth D>500 um.Furthermore, while higher concentration achieves larger peak stiffness, it also suffers a smaller crosslink depth, as shown by Eq. ( 17) and Fig. 10.Therefore optimal concentration range of 0.8% to 1.2% are recommended.It should be noted that the RF solution concentration is always less than the effective concentration diffused into the stroma per reported data.[40][41][42].

CONCLUSION
We have presented, for the first time, analytic formulas for the safety dose, minimum corneal thickness and RF concentration characterized by the extinction coefficient (A), concentration and the diffusion depth of the riboflavin, the UV light intensity, dose, irradiation duration (t), and the corneal thickness (z).The safety dose (E*) is an increasing function of the parameter set (D, z, C 0 ) and has a range of 5.3 to 10.1 J/cm 2 for cytotoxicity threshold 0.63 to 1.26 J/cm 2 .Minimum corneal thickness z* =(300, 400) um for safety dose of E*=(5.0,10.1) J/cm 2 which is much higher than the conventional dose 5.4 J/cm 2 (with z*=400 um).For maximum efficacy, the optimal dose is 0.7 to 1.5 J/cm 2 .However, to achieve crosslink depth of 230 to 300 um, higher dose of 2.0 to 3.0 J/cm 2 is recommended.It should be noted that the formulas presented in this study would require further justification by clinical data (for human corneas), particularly the measured value of the extinction coefficients of the photolysis product, the quantum yield and the cytotoxic threshold of the endothelium cells which are not yet available.

CONSENT
It is not applicable.

Fig. 1 .
Fig. 1.A corneal model system under UV light cross-linking for the epi-on (A) and epioff (B) case, where z is along the corneal thickness direction (z=0 for the corneal surface); the red dotted curves define the riboflavin diffusion boundaries[18] Eq. (5), we obtain the analytic formula for the RF concentration, from Eq. (2.b).

Fig. 5 .Fig. 6 .
Fig. 5.The safety corneal thickness (z*) versus concentration for E*/E 0 =1.0 and D =(200, 300, 500) um for curve (1,2,3) Fig.8shows S' versus z for a fixed exposure time t=100 s, but for various UV light intensity, where S* is a decreasing function of the UV light intensity (for a given exposure time), whereas Z* is an increasing function of I 0 (or dose E 0 =tI 0 ) which is shown in Fig.9and can be well fit by Eq. (9.b) with ag/M=0.07.By more simulation profiles for various D and C 0 (not shown here), we found Z* versus C 0 and D in Fig.10 and 11.Combining the curves shown by Figs. 9 to 11, we are able to figure out the following approximate analytic equations for S* and Z* for the linear regime (for dose between 2.0 and 3.0 J/cm 2 as shown in Fig.8).S*=10 -0.5(E-2.0)+ 62(C 0 -0.1) +0.005(D-500)(18.a)