Fig. 1 A corneal model system under UV light crosslinking for the epi-on (A) and epi-off (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.
Fig. 2 The normalized safety dose (E*/E0) versus C<sub>0</sub> for D=500 µm and z = (350, 400, 500) µm, for curve (1,2,3), where the red dot shows the reference point, or E*=E0.
Photopolymers have also been explored for uses in dentistry, drug delivery, tissue engineering and cell encapsulation systems3. Both ultraviolet, visible and infrared lights (360 to 1000 nm) have been used as the photoinitiators for various photosensitizers1.
For ophthalmology applications, corneal collagen cross linking (CXL) systems have been commercialized for years for human clinical uses2.
Photochemical kinetics of CXL and the biomechanical properties of corneal tissue after CXL are reported2.
However, much less efforts have been invested in basic theoretical studies of photopolymerization 3,4, where Lin presented the first dynamic modeling for the safety of CXL5-8. The safety and efficacy issues of CXL have been explored clinically and theoretically8,9
The critical issues of CXL to be explored in this paper are characterized by the following ten (10) key parameters: the 3 extinction coefficients (Aj, with j=1,2,3), concentration (C) and diffusion depth (D) of riboflavin, UV light dose (E), corneal thickness (z), and finally the cytotoxic energy threshold of endothelial cells (Ed). The dose is further defined by the product of the light intensity (I) and the exposure duration (t), i.e., E=It.
The extinction coefficients (Aj with j=1,2,3) are furtherdefined by three absorption constants of the UV light in riboflavin (without corneal stroma), in the photolysis product, and in the corneal stroma (without riboflavin).
Our theory will show that the safety dose (E*) and the corneal safe thickness (z*) cannot be set as a constant. Instead, they are variable functions defined by the combined parameter set (Aj,C,Ed,D). Finally, we will present various safety zones, defined by the parameter sets (z*,D*), (z*,C*) and (C*,D*), as the guidance for feature clinical studies. This study will use the currently available (or measured) data whereas the unknown parameters will be treated as free parameters within the clinically recognized ranges.
2.Methods and theory
2.1 The Modeling System
As shown in Figure 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 (approximately 0.1% to 0.25%) 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 Figure1) 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 efficient5.
2.2 The UV light intensity
In the above described eye model, the UV light intensity in the corneal stroma is given by a revised time-dependent Lambert-Beer law4-6
I(z,t)=I0exp[-A(t)z] (1)
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 be27 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+ε1C0), with the reported datA²8 ε1= 204 (%·cm)-1. For t>0, A(t) is an increasing function due to the deletion of C0 in time and defined by both the extinction coefficient of the RF(ε1) and its photolysis product (ε2) which is not yet available for human, but was estimated to be about 80 to 120 (%·cm)-1, based on measured data in RF solution under UV light irradiation 4,5.
2.3 The safety dose
The normalized safety dose (or fluence) applied to the cornea collagen (at a depth z) for an UV exposure time (t) is given by8
E* = (E’+gz) exp(A²z), (1.a)
A²=2.3[Q+ ε2C0G(z)] (1.b)
where gz is a correction factor for the transient state given by g=0.5(I2–I1)T/z, with T being the steadystate cross-linking time fit numerically to be T = 8/(aI0). G(z) =1- 0.25z/D, with D being the diffusion depth and E’ is the endothelium cells damage threshold based on z=400 um, D=200 um and C0 is 0.02%. As shown by Fig. 2 the normalized safety dose (E*/E0), is an increasing function of (zDC0), where E0 is the safety dose at the reference point of z=400 um, D=500 um and C0 = 0.02%, with =3.1 J/cm², for E’ =0.63 J/cm².
2.4 The cross linking time (T*)
The RF concentration is given by 7,8
C(z,t)=C0F(z)exp[-aE(z,t)] (3)
The cross linking time (T*) is defined by when
C(z,t=T*)/C0= 0.018 (or e-4), or aE=4, which leads to the formula for the cross linking time
T*(z,t)=T0(1+0.5agz)exp(A²z) (4)
T0 is the surface cross linking time given by T0=T*(z=0)=4000/(aI0), for I0 in mW/cm², and a=83.6λ∅ ε1= 6.2(∅ 0.1), with ∅ being the quantum yield.
For example, for I0 =10 mW/cm², we obtain T0=64.4 and 32.2 seconds, for quantum yield ∅ =0.1 and 0.2.
Solving Eq. (2) for C0, the safety (or minimum) concentration
C0*=[In(E*/(E' +gz))_32z]/(2.3ε2Gz) (5)
2.5 CXL Efficacy
The efficacy of CXL is further defined by the increase in corneal stiffness (S) after CXL given by (6)
Our numerical solutions (to be presented elsewhere) show the increase of the stiffness is given by the following scaling laws:
Our numerical solutions (to be presented elsewhere) show the increase of the stiffness is given by the following scaling laws:
For transient state: for a given dose E0.
For steady-state
For maximum stiffness (S*), for a fixed dose.
The scaling law for the optimal dose is given by
3.Clinical guidance and new protocol
Our above-presented theory provides us the new clinical guidance for CXL summarized as follows:
(a) The safety minimum corneal thickness (z*) is a decreasing function of the product of DC0. Therefore, for a given dose, thin corneas (350-390 µm) require high concentration and deep diffusion depth (or longer pre-operation RF diffusion duration). On the contrary, thick corneas (410-450 µm) only require small D or C0.
(b) Higher RF safety concentration is required for thinner cornea and smaller diffusion depth. For examples, the safety conditions are (for D=500 µm):
For D=500 µm, C0*= (0.04, 0.03, 0.02, 0.01)%, for z=(351, 375, 400, 425) µm.
4.Conclusions:
A new CXL protocol is suggested based on new safety criteria and optimal efficacy. Under the safety criteria thin cornea range of 350 to 390 um can be treated, whereas thick cornea of 450 to 500 um may not be safe in low concentration and/or small diffusion depth (D).
The conventional safety criteria of [E*=5.4 J/cm² and z>400um] and accelerated CXL using a linear theory are questionable. Our theory predicts some of the reported clinical feature for optimal stiffness using a lower dose.
