Phakic Toric IOL Optimal Rotation Calculator

Glossary – Terms Used

SE (Spherical Equivalent)

SE = S + C/2. Used to compare the achieved and the targeted values.

Spectacle plane / Corneal plane / IOL plane

Optical planes where powers are expressed. Conversions use the vertex distance and the ELP.

K1 / K2 and A1 / A2

Keratometric radii (K1 is the flattest, K2 the steepest) and their orthogonal axes (A2 = A1 + 90°).

PCA / Posterior Corneal Astigmatism

Contribution from the posterior surface of the cornea. Can be estimated using different methods (Koch/Baylor, Abulafia, etc.).

SIA

Surgically Induced Astigmatism: astigmatism induced by reopening the incision, entered as magnitude/axis.

Negative / Positive cylinder conventions

Same astigmatism expressed in C− or C+. Conversion: C+ = −C−, A+ = A− + 90°.

J0 / J45

Vector coordinates: J0 = −(C/2)·cos(2A), J45 = −(C/2)·sin(2A). Useful for vector addition of components (cornea, IOL, etc.).

ELP (Effective Lens Position) / Cornea–ICL distance

ELP stands for Effective Lens Position, i.e., the distance between the corneal vertex and the principal plane of the intraocular lens. It represents the optical separation used in all thin-lens equations.

In this phakic ICL setting, the "cornea–phakic IOL distance" plays the same optical role as an ELP in pseudophakic eyes: it is the distance between the corneal vertex and the principal plane of the additional lens. In practice it can be estimated from anterior segment OCT or Scheimpflug imaging (ACD + vault).

Using a realistic measured or estimated cornea–ICL distance allows a more accurate conversion of the toric ICL cylinder from the implant plane to the corneal plane (see Gatinel conversion method – Q ratio).

Ps / Pc (Phakic ICL)

Ps and Pc refer respectively to the spherical power and the cylindrical power of the phakic toric ICL, both defined at the ICL plane.

For myopic correction, Ps is typically negative (e.g., −10 D). The Pc term corresponds to the toric or cylindrical component of the ICL (often positive on the label, e.g., +2.00 D), whose effective contribution at the corneal plane depends on the cornea–ICL distance and on the selected conversion model (e.g. vergence method or Gatinel Q-ratio).

Gatinel conversion method (Q ratio)

Q is the local ratio Q = ∂P/∂R linking a change in IOL power dP to the induced change in ocular refraction at the corneal plane dR (signs are opposite: increasing IOL power makes refraction more myopic).

In a paraxial eye model with an additional intraocular lens (phakic ICL or pseudophakic add‑on):
Q = − 1 / [ 1 − (E / na) · (K + R) ]2, where E is the cornea–phakic IOL distance (meters), na = 1.336, K the keratometric power, and R the refraction at the corneal vertex (corneal plane).
Hence dR = (1/Q)·dP.

Toric conversion. Compute Q on the two principal meridians (K1, K2) and average: Pc,eff ≈ ½ · ( |1/Q(K1)| + |1/Q(K2)| ) · Pc. This is what the solver uses to convert a toric IOL cylinder at the IOL plane to its effective cylinder at the corneal plane.

Versus classic “vergence” translation. The thin‑lens translation Feff = F / (1 + (E/na)·F) ignores corneal power. The Q‑based approach explicitly includes K (via K+R) and ELP, so it better captures extremes (flat/steep corneas, anterior/posterior ELP).

Typical magnitudes (R=0): for K≈43 D and ELP≈3.5–5.0 mm, |Q|≈1.3–1.5 so |1/Q|≈0.65–0.77 per diopter. See the tables for a full grid of values.

Reference : Gatinel D, Debellemanière G, Saad A, Malet J. Relationship between intraocular lens power and ocular refraction variations. J Cataract Refract Surg. 2025 Jun 1;51(6):488-495.

θ_Cmin / θ_Ctarget

Rotations producing the minimal cylinder and the minimal deviation from target (SE/J0/J45), respectively.

ΔSE / ΔC

Differences between the expected post-rotation refraction and the measured (or targeted) refraction, at the spectacle plane.

Cornea–phakic IOL distance

Axial distance between the anterior corneal vertex and the principal plane of the phakic toric IOL (ICL) in the ciliary sulcus. It is the key parameter used to convert powers between the ICL plane and the corneal plane in both the vergence‑based model and the Gatinel Q‑ratio model.

Subsidiary astigmatism

Residual refractive astigmatism component that cannot be fully corrected by the chosen toric phakic ICL—either because the available cylinder power does not exactly match the total refractive astigmatism (corneal + internal), or because the toric axis cannot be perfectly aligned. This value influences the predicted residual cylinder after optimal rotation.

Ideal ICL

Theoretical total toric phakic ICL power (sphere and cylinder at the ICL plane) that would achieve the target refraction (emmetropia if target is 0). Calculated using the Gatinel Q-ratio formalism: the sphere is derived from ΔR (corneal plane) with dP = −Q × ΔR; the cylinder is computed vectorially as J_target − (J_cornea + J_internal), then converted from corneal plane to ICL nominal power. The ΔPs value indicates the difference vs. the implanted ICL sphere.

Optimal zone

Range of rotations where the refractive cylinder remains ≤ the chosen threshold (e.g., 0.75 D).

Toric Add-on IOL (Piggyback)

This calculator can also be used for toric sulcus add-on IOLs in pseudophakic eyes. Treat the add-on as a phakic IOL and enter the cornea–add-on distance as the ELP (typically 3.0–3.5 mm). The vectorial analysis and rotation optimization apply identically.