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Reducing the diameter of UV-A beam in EPI-ON Custom Fast Cross-linking (CFXL), a pachymetry dependent fluence, lower power, shorter duration treatment: some mathematical considerations.
by G. Barbaro, C. Caruso
The cross-linking treatment should provide outstanding improvements of the corneal curvature when the less curved portions of the cornea are not exposed to the UV-A shaft for irradiating the cornea. A desired correction of the shape of the keratoconic cornea should take place into few hours from the end of the cross-linking treatment.
Let us assume that a tiny portion of a shell having the form of a surface of revolution is strengthened. With a cross-linking treatment, such a situation can occur only on the tiny portion. The equations describing the equilibrium of forces and torques on this infinitesimal portion of the shell are given by the theory of elastic shells:[1,2]
Stretching in correspondence of the intermediate surface of the shell is described by the following equations:
Eliminating w from the above equations, we obtain
Assuming that the displacement v of the treated portion of the cornea is substantially negligible along the direction of the meridian, since the treated portion is small in respect to the surrounding portion, it results that the right side of the above equation should remain practically constant during the treatment.
The strain components can be expressed in terms of the stress forces by applying the Hooke’s law:
thus an increase of the Young’s modulus E causes a reduction of the strain components.
In corneas affected by keratoconus, the apical portion of the cornea is characterized by a relatively small radius of curvature. In the above figure, the keratoconic portion of the cornea is bordered by the latitudinal line at the angle j. The surrounding portion is typically straighter than in a normal cornea, and exerts a stress force and a bending moment on the apical portion along the longitudinal direction of the angle j.
Given that the surrounding portion has not been treated and that the treated portion is tiny in respect to the untreated portion, it may be assumed that the values of the longitudinal stress Nj that the surrounding portion of the shell exerts on the treated portion are substantially unchanged. Because of the crosslinking treatment, Young’s modulus E of the treated portion, have increased. Therefore, the strain components ej and eq of the treated portion of the cornea are decreased.
Therefore, since the term
should remain constant for the reasons stated above, it results that the radius of curvature r1 should increase for compensating the reduction of the strain components.
From the above mathematical reasoning, it is expected an increase of the radius of curvature r1 in correspondence of the latitudinal line at the angle j for compensating the increase of the Young’s modulus E. In other words, by treating locally a small portion of a cornea and leaving untreated the remaining part it is expected a local increase of the radius of curvature r1, that is a local flattening of the cornea.
If the radius of curvature r1 of the treated portion increases whilst the value of the stress forces on the infinitesimal portion remains substantially unchanged, then the stress forces that the infinitesimal portion exerts on the surrounding untreated portion of the cornea have substantially the same value but have rotated in order to remain tangent to the longitudinal line: this rotation is expected to cause a bending of the surrounding portion.
It is expected that strengthening locally a small portion of a shell, the small portion having a smaller radius of curvature than the radius of curvature of the surrounding portion, cause a flattening of the treated portion and a bending of the surrounding portion.
With a cross-linking treatment with a reduced diameter of the UV-A shaft, for example, from 3 to 5 mm, the area in which keratoconus is usually located, carried out on the most curved corneal area, a rapid flattening, along with a more rapid flexion of the untreated portion, should take place.
The curve sketched in the above figure schematically represents a cross-sectional profile of a cornea, tied to the sclera in correspondence of the coordinates a and b. A number N of sectors of a same width of the cornea are labeled S1, …, Si, …, SN. A generic sector Si spans between the coordinates xi-1 and xi.
There is a mathematical condition that is always satisfied: all points of the cornea y(x) are defined only for x ranging between a and b. This means that the whole corneal surface is always comprised between the vertical straight lines passing through the points at which the cornea is tied to the sclera.
It will be demonstrated that an increase of curvature in a generic sector Si causes a decrease of curvature in at least another different sector.
The curvature C(x) of a curve y(x) at the abscissa x is given by the following equation:
The average curvature ACi of the curve y(x) in the generic sector Si is:
Since the N sectors have the same width and are comprised between a and b,
Therefore, the sum of the average curvatures of all sectors is:
Since the whole corneal surface is always comprised between the vertical straight lines passing through the points at which the cornea is tied to the sclera, then at coordinates a and b the curve y(x) cannot be directed outwards the interval [a,b]. This means that
thus the sum of the average curvatures of all sectors cannot exceed a maximum value:
Therefore, if the average curvature of a generic sector Si increases, as a general rule a reduction of the average curvature of at least a different sector is necessary to satisfy the above inequality. For example, if the average curvature ACi of a generic sector Si approaches the limit value
then the average curvature of all other sectors must nullify, i.e. all other sectors must become substantially straight, in order to comply with the above inequality.
In numerous cases, it is observed that the slope of a perimetral portion of the cornea directly tied to the sclera remains substantially unchanged during the cross-linking treatment. Then at coordinates a and b the slope of the curve y(x) remains constant, thus
If the sum of the average curvatures remains constant, then any increase of the average curvature of a sector must necessarily be accompanied by a reduction of the average curvature of another sector.
 Timoshenko S, Woinowski-Krieger S. Theory of Plates and Shells. 2nd ed. McGraw Hill; 1959
 Timoshenko S, Goodier JN. Theory of Elasticity. 2nd ed. McGraw Hill; 1951.