J. Eur. Opt. Society-Rapid Publ. 22, 24( 2026) 247
The femtosecond laser cavitation bubble expansion and collapse remain highly confined compared to the overall treatment diameter, which can be several millimetres, which justifies the local geometric modelling assumptions employed in this work.
2 Material and methods
2.1 Calculation of the overall dose per treatment
The dose, or total fluence, in refractive laser surgery is defined as the average energy delivered per unit surface area of the treated corneal region. Because each laser pulse within a given cut is delivered with a constant energy level( though this may vary between cuts within a treatment), the dose effectively reflects the pulse density, that is, the number of pulses delivered per unit area.
The following equation applies to the calculation of the overall dose per treatment D T [ 26 ]:
D T ¼ E T
A T
;
D T ¼ n E p; ð2Þ p R 2 T
where E T is the total energy deposited during a treatment, E P is the energy of a single pulse, A T is the area of the treatment and R T its radius. The number“ n” represents the number of pulses assuming that all pulses carry the same energy, and pulses are approximately evenly distributed in the treatment area.
The total number of pulses delivered can be estimated geometrically. Assuming pulses are distributed in concentric circular paths( laps), the total number of laps and the average lap radius can be used to calculate the path length. This path length when divided by the spot distance gives the total number of pulses( for a single pass over the treatment area) as follows [ 26 ]:
n ¼ p R 2 T Spot Distance Track Distance;
where Spot Distances and Track Distance are distances between deposited neighboring spots along two essentially perpendicular axes( e. g., x and y, but also r and h).
Therewith, the overall dose per treatment simplifies to:
E P
D T ¼
Spot Distance Track Distance: ð4Þ
This expression shows that the Dose is independent of treatment area and is solely a function of pulse energy and spatial pulse density. The product of Spot Distance Track Distance can be interpreted as the effective area assigned to each pulse, rather than the physical bubble size itself.
On the other hand, the lower limit( minimum) dose( D Min) per treatment for a bridge-free dissection can be equated as: ð1Þ ð3Þ
D Min ¼
E P
Bubble Area ¼
E P
: ð5Þ p R 2 B
This expression shows that the minimum dose is a function of pulse energy and bubble area, i. e., limited by the physical bubble size itself.
In this work, the terms“ Bubble Area” and“ Bubble Diameter” actually refer to the tissue disruption size( area or diameter) caused by a single bubble. It is not a timedependent quantity since it refers to the effective size once the bubble completes the disruption work, thus it requires the bubble already collapsed in time after creating the disruption effect. Therefore, the Bubble Area refers to the planar projection of the cavitation bubble onto the cut surface, not the total bubble surface. In a 3D lattice, each Voronoi cell is formed by 4 or 6 neighboring spots, which occupy a very small spatial window relative to corneal curvature. Hence, the cornea is locally planar in 3D over each cell [ 24 ]. Therefore, local curvature effects are negligible at the scale of individual Voronoi cells, and global corneal curvature does not influence the local overlap statistics used in the model, justifying that scanning pattern geometry( spiral, raster, etc.) does not alter the local overlap physics, allowing the problem to be treated in a locally Cartesian framework.
2.2 Calculation of the overall dose per treatment as a function of scaling factors for Spot and Track Distance
Considering a scaling factor‘ F’ that is dimensionless and affects spacing relative to bubble size, one can calculate the Spot and track distance with respect to Bubble Diameter as follows:
Spot Distance ¼
Bubble Diameter; ð6Þ
F1
Bubble Diameter Track Distance ¼; ð7Þ
F2
where F1 and F2 are scaling factors for Spot and Track Distance, respectively, and in general, values F1 orF2 > 1 represent effective spot or track overlap, and values F1 orF2 < 1 actually suggest disjoint spot or track cumulation. Therefore, if F1 and F2 are large, the Spot and Track Distances would be a small fraction of Bubble Diameter, resulting in more overlap, a higher number of pulses per area, and a higher dose, resulting potentially in a clean cut. Conversely, if F1andF2are small, the Spot and Track Distances may become large multiples of Bubble Diameter, resulting in less to no overlap, lower dose, but may pose a higher risk of incomplete cut.
Using equation( 4), and applying equations( 6) and( 7), D T can be expressed as follows:
D T ¼
E P F1 F2 Bubble Diameter 2 ð8Þ