Matrix-based calculation scheme for toric intraocular lenses*

Abstract Background and purpose: While a number of intraocular lens power prediction formulas are well established for determination of spherical lenses, no common strategy is published for the computation of toric intraocular lenses. The purpose of this study is to describe a paraxial computing scheme using 4 × 4 system matrices to describe the ‘optical system eye’ containing astigmatic refractive surfaces with their axes at random. Methods: Based on the definition of a centred optical system in the paraxial Gaussian space containing astigmatic surfaces using 4 × 4 refraction and translation matrices, we derived a methodology for calculating the refractive power of thin and thick toric intraocular lenses by solving a linear equation system. In a second step, we derived a methodology for prediction of the residual spectacle refraction after implantation of any toric lens implant with any orientation. Results: The capabilities of this computing scheme are demonstrated with three examples. In example 1 we calculate a ‘thin toric lens’ for compensation of a corneal astigmatism to achieve a spherical target refraction. In example 2 we compute a ‘thick toric lens’, which has to compensate for an oblique corneal astigmatism and rotate the spectacle cylinder to the ‘against the rule’ position to enhance near vision. In example 3 we predict the residual refraction at the corneal plane after implantation of a thick toric lens, when the cylinder of the lens implant is compensating the corneal cylinder in part and the axis of implantation is not fully aligned with the axis of the corneal astigmatism. Conclusion: We present an en bloc matrix‐based strategy for the calculation of thick or thin toric intraocular lenses, with the flexibility of crossing an unlimited number of cylinders with restrictions to paraxial optics. The resulting system matrix S is written as a product of 4 × 4 refraction and translation matrices. Residual refraction at the corneal (contact lens) or spectacle plane can be derived by inverting the order of matrices for calculation of the system matrix.